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24
mcap
standalone
maths
Geometry
The figure shows the dimensions of a bottle of shampoo. The cap at the top of the shampoo bottle can be modeled by a cylinder with a diameter of 1 inch and a height of 1 inch. A student wants to mail 2 bottles of shampoo to a friend. The student packs the 2 bottles of shampoo in a box that is 6 inches long, 4 inches w...
null
How much packing material, in cubic inches, will the student need to fill the empty space in the box? Show your work or explain your answer.
Volume of the box: $6 \times 4 \times 10 = 240$ cubic inches. Volume of the bottle: $\pi \left(\frac{2.4}{2}\right)^2 (7) + \pi \left(\frac{1}{2}\right)^2 (1) = 10.33\pi$ cubic inches. Empty space in the box: $240 - 2 \times (10.33\pi) \approx 175.09$ cubic inches. A total of 175.09 cubic inches of packing mat...
train
calculation_with_work
context_first_then_domain_knowledge
25
mcap
standalone
maths
Geometry
The coordinates of the vertices of quadrilateral $PQRS$ are shown in the $xy$ -plane. ![A coordinate plane showing quadrilateral PQRS. The vertices are P(4, 10), S(4, 5), R(6, 4), and Q(10, 7). The x and y axes range from 0 to 10.]
null
Prove that quadrilateral $PQRS$ is a trapezoid. Determine whether quadrilateral $PQRS$ is an isosceles trapezoid. Show your work or explain how you determined your answers.
Calculating the slopes: The slope of side PQ: $\frac{10-7}{4-10} = \frac{3}{-6} = -\frac{1}{2}$ The slope of side RS: $\frac{5-4}{4-6} = \frac{1}{-2}$ The slope of side PS: $\frac{10-5}{4-4} = \frac{5}{0}$ , which is undefined The slope of side QR: $\frac{7-4}{10-6} = \frac{3}{4}$ $PQRS$ is a trapezoid be...
train
calculation_with_work
context_first_then_domain_knowledge
26
mcap
standalone
maths
Geometry
A student is standing next to a vertical flagpole. The top of the student's shadow coincides with the top of the flagpole's shadow as shown. ![Diagram showing a student and a vertical flagpole standing next to each other, with the sun shining from the top right. The shadows of both the student and the flagpole coincid...
null
Based on the given information, what are the **least** and **greatest** possible heights, in **feet**, of the flagpole? Explain how you arrived at your answers.
The least height for the flagpole will be when the student's shadow is longest (7.5 feet or 90 inches) and the distance between the student and the flagpole is shortest (21 feet or 252 inches). The following proportion can be solved to arrive at the least height of the flagpole, in inches. $\frac{62}{90} = \frac{x}{9...
train
calculation_with_work
context_first_then_domain_knowledge
40
new_york_regents
standalone
biology
Genetics
**Prostaglandins: One of the Most Important Molecules You Never Heard of** When cells experience trauma and are damaged, they quickly communicate this damage to nearby cells. One way this is accomplished is by the release of chemical messengers known as prostaglandins. Prostaglandins signal cells to respond with inflam...
null
Describe a qualitative or quantitative constraint that these researchers took into account when using animal models in the design of their experiment. [1]
["Using humans in the early stages of research raises ethical concerns.", "The DNA of mice must be similar to human DNA, so the response of mice to this cancer research could be applied to humans.", "It would cost significantly less to work with mice than with humans, so more mice could be used in each trial, increasin...
train
brief_explanation
context_first_then_domain_knowledge
54
pennsylvania_keystone
standalone
chemistry
kinetics
Rolando drives at least 40 miles but less than 60 miles each week.
null
Graph the compound inequality representing all of the possible distances Rolando could drive for 8 weeks.
Number line showing Miles Driven from 0 to 600, marked every 40 units. A closed circle is placed at 320, and an open circle is placed at 480. The segment between 320 and 480 is shaded.
train
brief_explanation
context_first_then_domain_knowledge
55
pennsylvania_keystone
standalone
chemistry
analytical_lab_methods
Points J and K lie on the same line, as shown on the coordinate plane below.
null
A. What is the slope of the line passing through points J and K? Show or explain all your work.
$\frac{1}{2}$
train
numerical_value
context_first_then_domain_knowledge
56
pennsylvania_keystone
standalone
chemistry
atomic_structure_periodicity
Points J and K lie on the same line, as shown on the coordinate plane below: A coordinate plane showing a straight line passing through points J(3, 3.5) and K(6, 5). The line extends into the second and fourth quadrants.
null
What is the slope of the line passing through points J and K? Show or explain all your work.
The slope is $\frac{1}{2}$.
train
numerical_value
context_first_then_domain_knowledge
57
pennsylvania_keystone
standalone
chemistry
atomic_structure_periodicity
Points J and K lie on the same line, as shown on the coordinate plane below: A coordinate plane showing a straight line passing through points J(3, 3.5) and K(6, 5). The line extends into the second and fourth quadrants.
null
Write the equation of the line passing through points J and K. Show or explain all your work.
The equation of the line is $y = \frac{1}{2}x + 2$.
train
equation_or_expression
context_first_then_domain_knowledge
58
pennsylvania_keystone
standalone
chemistry
solutions_mixtures
Points L and M are added to the coordinate plane. The slope of $\overleftrightarrow{JK}$ is equal to the slope of $\overleftrightarrow{LM}$.
null
Describe two ways the lines could be related.
The lines could be parallel or the lines could be the same line (collinear).
train
short_answer
context_first_then_domain_knowledge
59
pennsylvania_keystone
standalone
chemistry
solutions_mixtures
Both the box-and-whisker plot and the histogram shown below represent the heights, in inches, of the same group of basketball players.
null
Based on the two data displays, what is the range of the heights?
The range is 14 inches.
train
numerical_value_with_units
context_first_then_domain_knowledge
60
pennsylvania_keystone
standalone
chemistry
acids_bases
Both the box-and-whisker plot and the histogram shown below represent the heights, in inches, of the same group of basketball players: Box-and-whisker plot showing heights of basketball players. The number line ranges from 72 to 88 inches. The minimum value is 72, Q1 is 78, the median (Q2) is 80, Q3 is 83, and the maxi...
null
Based on the two data displays, what is the interquartile range of the heights?
The interquartile range is 6 inches.
train
numerical_value_with_units
context_first_then_domain_knowledge
61
pennsylvania_keystone
standalone
chemistry
acids_bases
Both the box-and-whisker plot and the histogram shown below represent the heights, in inches, of the same group of basketball players: Box-and-whisker plot showing heights of basketball players. The number line ranges from 72 to 88 inches. The minimum value is 72, Q1 is 78, the median (Q2) is 80, Q3 is 83, and the maxi...
null
Based on the two data displays, how many of the basketball players are 78 inches tall?
3 basketball players are 78 inches tall.
train
numerical_value_with_units
context_first_then_domain_knowledge
62
pennsylvania_keystone
standalone
chemistry
acids_bases
Both the box-and-whisker plot and the histogram shown below represent the heights, in inches, of the same group of basketball players: Box-and-whisker plot showing heights of basketball players. The number line ranges from 72 to 88 inches. The minimum value is 72, Q1 is 78, the median (Q2) is 80, Q3 is 83, and the maxi...
null
Based on the two data displays, what is the mean of the heights?
The mean of the heights is 79.25 inches.
train
numerical_value_with_units
context_first_then_domain_knowledge
63
pennsylvania_keystone
standalone
chemistry
bonding_structure
Lucy randomly surveyed 25 students in her school about the number of bottles of water they consumed in the last week. She recorded the results in the bar graph shown below.
null
A. What is the mode of the responses from Lucy's survey?
0
train
numerical_value
context_first_then_domain_knowledge
64
pennsylvania_keystone
standalone
chemistry
equilibrium
Lucy randomly surveyed 25 students in her school about the number of bottles of water they consumed in the last week. She recorded the results in the bar graph shown below.
null
B. How many bottles of water, in all, did the 25 students Lucy surveyed consume last week?
35
train
numerical_value
context_first_then_domain_knowledge
65
pennsylvania_keystone
standalone
chemistry
equilibrium
Lucy randomly surveyed 25 students in her school about the number of bottles of water they consumed in the last week. She recorded the results in the bar graph shown below.There are 236 students in Lucy�s school
null
C. Based on her graph, how many of the 236 students should Lucy expect to have consumed either 0 bottles or 1 bottle of water last week?
151
train
numerical_value
context_first_then_domain_knowledge
66
pennsylvania_keystone
standalone
chemistry
solutions_mixtures
Lucy randomly surveyed 25 students in her school about the number of bottles of water they consumed in the last week. She recorded the results in the bar graph shown below.Lucy determines that the interquartile range of the 25 responses is q
null
D. Write an expression to represent the interquartile range of the responses Lucy should expect to get, based on her graph, if she surveys all 236 students in her school.
q
train
equation_or_expression
context_first_then_domain_knowledge
94
pennsylvania_keystone
standalone
chemistry
acids_bases
The map shows some information about two ecosystems. Location of Two Types of Ecosystems![Map showing the location of two ecosystems: a rainforest in Belize (Central America) and the Sahara Desert (North Africa). The map includes latitude lines (60°N, 40°N, 20°N, 0°, 20°S, 40°S) and a compass rose.](a13813eefbf1c8dfb8...
null
Describe a relationship between a biotic component and an abiotic component in either a desert or a rainforest ecosystem.
Ecosystem: Desert. Relationship: The sand (abiotic) in deserts provides camouflage for some species of spiders and snakes (biotic) so they can capture their prey easier.
train
brief_explanation
context_first_then_domain_knowledge
95
pennsylvania_keystone
standalone
chemistry
kinetics
A model is shown.
null
Identify the maximum number of bonds one carbon atom can form.
One carbon atom can form a maximum of four bonds.
train
short_answer
context_first_then_domain_knowledge
96
pennsylvania_keystone
standalone
chemistry
kinetics
A model is shown.
null
Describe how electrons are involved in the formation of biological macromolecules.
Electrons are involved in forming covalent bonds between atoms, and covalent bonds are required to form biological macromolecules.
train
brief_explanation
context_first_then_domain_knowledge
97
pennsylvania_keystone
standalone
chemistry
kinetics
A model is shown.
null
Carbon is often considered the backbone of biological macromolecules. However, hydrogen is the most abundant element in the universe. Explain why the backbone of large macromolecules is made of carbon atoms rather than hydrogen atoms.
Because carbon atoms make four covalent bonds, they can form many types of backbones with many other atoms attached. Hydrogen can only make one covalent bond, so it cannot form chains or backbones.
train
brief_explanation
context_first_then_domain_knowledge
98
pennsylvania_keystone
standalone
chemistry
kinetics
The map shows some information about two ecosystems.
null
Describe an abiotic characteristic that is similar in both the desert and the rainforest.
The presence of soil is an abiotic characteristic that is similar in both a desert and a rainforest
train
brief_explanation
context_first_then_domain_knowledge
99
pennsylvania_keystone
standalone
chemistry
kinetics
The map shows some information about two ecosystems.
null
Describe a biotic characteristic that is unique to either a desert or a rainforest ecosystem.
A biotic characteristic that is unique to a rainforest ecosystem is the presence of bromeliads
train
brief_explanation
context_first_then_domain_knowledge
100
pennsylvania_keystone
standalone
chemistry
redox_electrochemistry
The map shows some information about two ecosystems.
null
Describe a relationship between a biotic component and an abiotic component in either a desert or a rainforest ecosystem.
Water is an abiotic component that must be ingested by all animals in either a desert or rainforest ecosystem
train
brief_explanation
context_first_then_domain_knowledge
101
pennsylvania_keystone
standalone
chemistry
redox_electrochemistry
Use the graph and diagram below to answer the question.
null
Explain how Enzyme A acts as a catalyst in the reaction. Be sure to include energy and time in your answer.
Enzyme A acts as a catalyst by reducing the activation energy
train
brief_explanation
context_first_then_domain_knowledge
102
pennsylvania_keystone
standalone
chemistry
redox_electrochemistry
Use the graph and diagram below to answer the question.
null
Conditions around an enzyme change and affect the shape of the enzyme's active sites. Predict how this would affect the enzyme's ability to catalyze the reaction.
When the shape of an enzyme's active site is changed, the substrate cannot attach to the active site; it will not "fit." The enzyme would not be able to catalyze the reaction. When the shape of the enzyme's active site is slightly changed (caused by a change in pH, for example), the enzyme activity can become greatly r...
train
brief_explanation
context_first_then_domain_knowledge
103
pennsylvania_keystone
standalone
chemistry
redox_electrochemistry
A student draws a Venn diagram to compare and contrast two cellular processes.
null
Select two areas from the Venn diagram and describe a characteristic that could be placed into each of the areas.
Area: X Characteristic: converts light energy from the sun into chemical energy in the form of glucose. Area: Z Characteristic: occurs in the mitochondria of eukaryotic cells
train
short_answer
context_first_then_domain_knowledge
104
pennsylvania_keystone
standalone
chemistry
redox_electrochemistry
A student draws a Venn diagram to compare and contrast two cellular processes.
null
Explain why some organisms would need to perform both cellular respiration and photosynthesis.
Some organisms would need to perform both cellular respiration and photosynthesis because photosynthesis is the only way those organisms could obtain glucose, which is broken down during cellular respiration in order to make ATP, which is necessary for all cells and is not created in great enough quantities during phot...
train
brief_explanation
context_first_then_domain_knowledge
105
pennsylvania_keystone
standalone
chemistry
redox_electrochemistry
The model shows a biochemical reaction utilizing sucrase.
null
Explain the role of sucrase in the reaction shown.
Sucrase is an enzyme that reduces the amount of energy (activation energy) needed to break down sucrose, a disaccharide, into glucose and fructose, two monosaccharides.
train
brief_explanation
context_first_then_domain_knowledge
118
staar
standalone
maths
Algebra
A paper airplane was thrown from the top of a tall building. The height of the paper airplane above the ground can be found using the function y = -1.5x + 60, where x is the time in seconds the airplane has been in the air.
null
How many seconds did it take the paper airplane to reach the ground?
40
train
numerical_value
context_first_then_domain_knowledge
119
staar
standalone
maths
Algebra
The graph of linear function $h$ is shown on the grid.
null
Given $f(x) = x$ and $h(x) = af(x)$, what is the value of $a$?
0.8
train
numerical_value
context_first_then_domain_knowledge
121
virginia_sol
standalone
maths
Algebra
The graph of $y = -x^2 - 2x + 8$ is shown.
null
On the grid, identify each of the solutions to $-x^2 - 2x + 8 = 0$.
Both of these points, and only these points, must be plotted on the coordinate plane: $(-4,0)$ and $(2,0)$.
train
brief_explanation
context_first_then_domain_knowledge
155
AP
final
biology
Cell Biology
Most proteins that are secreted from a cell must be transported to the endoplasmic reticulum (ER) either during translation or after translation. For proteins transported during translation, this process begins in the cytosol and pauses when a specific sequence of amino acids is translated. The translation complex is t...
null
C. iii. Protein 1 is encoded by 234 nucleotides, while protein 2 is encoded by 495 nucleotides. Assuming all nucleotides for both proteins encode amino acids, calculate the difference in the number of amino acids between the two proteins.
87 (amino acids) $[(495/3)-(234/3)]$
train
equation_or_expression
context_first_then_domain_knowledge
218
AP
final
chemistry
equilibrium
$$C_3H_6O_3(aq) + NaOH(aq) ightarrow NaC_3H_5O_3(aq) + H_2O(l)$$ A student is studying the reaction between lactic acid, $C_3H_6O_3$, and sodium hydroxide, NaOH, as represented in the balanced equation above. In a third experiment, the student investigates the enthalpy of the reaction between lactic acid and sodium hy...
null
Calculate the quantity of heat produced in the reaction, in J.
$$q = mc\Delta T = (200.0 \text{ g})(4.2 \text{ J}/(\text{g} \cdot ^\circ\text{C}))(23.2^\circ\text{C} - 20.0^\circ\text{C}) = 2700 \text{ J}$$
train
equation_or_expression
context_first_then_domain_knowledge
274
AP
final
EnvironmentalScience
earth_systems_resources
Urban areas have a heavy dependence on automobiles and often experience photochemical smog.
Satellite Track and Water Temperature Log of the Movement of One Adult Manatee Over a 12-Hour Period
The diagram shows the movement of one adult manatee over a 12-hour period across an active shipping channel during the winter. Manatees are large, aquatic, herbivorous mammals that primarily eat seagrass. Manatees can travel several miles to graze. Manatees mature slowly, have low biotic potential, and inhabit warm coa...
Sunlight is more intense/there is more sunlight in summer, and more primary pollutants/nitrogen oxides/<math>NO_x</math> react in sunlight. Summertime temperatures are warmer than other seasons, and chemical reactions occur more quickly in warmer temperatures. Trees release more VOCs in summer leading to more reactions...
train
brief_explanation
context_first_then_domain_knowledge
366
AP
final
history
world_history
Document 1 Source: S. M. Prokhorov, Soviet artist, *Students from a Workers' School*, poster for public display, 1928 ![Poster titled 'Students from a Workers' School' by S. M. Prokhorov, 1928. It depicts a line of diverse workers (men and women, dressed in working clothes and traditional attire) carrying books and p...
null
In the period circa 1750–1900, societies across the globe were affected by new technologies that transformed methods of production. Develop an argument that evaluates the extent to which the growth of industrialization led to economic or social change during this period.
From 1750 to 1900, new inventions such as the steam engine and mechanized textile machines expanded factory production. Industrialization caused major economic and social change by increasing production, growing cities, and creating new social classes. Factories replaced much home-based production and drew workers fro...
train
long_explanation
context_first_then_domain_knowledge
392
AP
final
physics
electricity_magnetism
A circuit consists of an ideal battery of emf $\mathcal{E}$ and four identical resistors $R_1$, $R_2$, $R_3$, and $R_4$, each of resistance $R$.
Figure 1: An ideal battery of emf $\mathcal{E}$ is in series with $R_1$. The current then divides between a branch containing $R_2$ and a branch containing $R_3$ and $R_4$ in series.
Derive an expression for the current $I_3$ in Resistor $R_3$.
$$I_1=I_2+I_3$$ $$I_2R=I_3(2R)$$ $$I_2=2I_3$$ $$I_1=3I_3$$ $$I_3=\frac{I_1}{3}=\frac{\mathcal{E}}{5R}$$
train
calculation_with_work
context_first_then_domain_knowledge
499
olympiad
final
chemistry
solutions_mixtures
The plot shows the energy of two hydrogen atoms as a function of their internuclear separation $r$.
Potential-energy curve for two hydrogen atoms as a function of internuclear separation, with a minimum near 74 pm at approximately $-432\ \text{kJ mol}^{-1}$.
Calculate the longest wavelength of light that has sufficient energy to break a H–H bond.
$\text{BDE} = 432 \text{ kJ mol}^{-1} = 7.17 \times 10^{-19} \text{ J/molecule}$ $7.17 \times 10^{-19} \text{ J} = hc/\lambda$, so $$\lambda = \frac{(6.626 \times 10^{-34} \text{ J s})(2.998 \times 10^8 \text{ m s}^{-1})}{7.17 \times 10^{-19} \text{ J}}$$ $$\lambda = 2.77 \times 10^{-7} \text{ m} = 277 \text{ nm}$$
train
calculation_with_work
context_first_then_domain_knowledge
500
olympiad
final
chemistry
solutions_mixtures
The plot shows the energy of two hydrogen atoms as a function of their internuclear separation $r$. The vibrational frequency of $\text{H}_2(g)$ is $1.32 \times 10^{14}$ Hz.
Potential-energy curve for two hydrogen atoms as a function of internuclear separation, with a minimum near 74 pm at approximately $-432\ \text{kJ mol}^{-1}$.
What percentage of the H–H bond dissociation energy is required to excite a molecular vibration of $\text{H}_2$?
$E = h\nu = (6.626 \times 10^{-34} \text{ J s})(1.32 \times 10^{14} \text{ s}^{-1}) = 8.75 \times 10^{-20} \text{ J}$. This is $$\left\{\frac{8.75 \times 10^{-20} \text{ J}}{7.17 \times 10^{-19} \text{ J}}\right\}\cdot100\% = 12.2\%$$ of the BDE.
train
short_answer
context_first_then_domain_knowledge
501
olympiad
final
chemistry
stoichiometry
Silicon is an industrially important semiconductor with a cubic unit cell in which Si atoms are represented as black spheres.
Cubic unit cell of silicon with Si atoms represented as black spheres.
How many Si atoms are contained in this unit cell?
There are 8 Si atoms in the unit cell.
train
calculation_with_work
context_first_then_domain_knowledge
502
olympiad
final
chemistry
states_of_matter_imfs
Silicon is an industrially important semiconductor with a cubic unit cell in which Si atoms are represented as black spheres.
Cubic unit cell of silicon with Si atoms represented as black spheres.
The length of the edge of the unit cell is 0.543 nm. Calculate the density of silicon in $\text{g cm}^{-3}$.
$$V=(0.543\times10^{-7}\text{ cm})^3=1.60\times10^{-22}\text{ cm}^3$$ $$\text{mass}=8\times\frac{28.09\text{ g mol}^{-1}}{6.022\times10^{23}\text{ mol}^{-1}}=3.73\times10^{-22}\text{ g}$$ $$\text{density}=\frac{3.73\times10^{-22}\text{ g}}{1.60\times10^{-22}\text{ cm}^3}=2.33\text{ g cm}^{-3}$$
train
brief_explanation
context_first_then_domain_knowledge
503
olympiad
final
chemistry
analytical_lab_methods
The amount of chloride in an unknown sample is determined by potentiometric titration using two silver/silver-ion half-cells connected by a salt bridge. A 0.200 M $AgNO_3$ titrant is added to the chloride-containing analyte in half-cell B; half-cell A contains 1.0 M $Ag^+$.
Schematic of the potentiometric titration cell, including the two Ag electrodes, the salt bridge, and the AgNO3 buret over half-cell B.
During the titration, when some of the $0.200 M$ $AgNO_3$ solution has been added to the analyte, which half-cell (A or B) contains the anode of the electrochemical cell? Explain your reasoning.
The concentration of $\text{Ag}^+$ ion will always be smaller in half-cell B than it is in half-cell A (it cannot exceed $0.2 M$!). Oxidation of the Ag electrode will take place in the half-cell with lower $\text{Ag}^+$ concentration, so the anode is in half-cell B.
train
calculation_with_work
context_first_then_domain_knowledge
504
olympiad
final
chemistry
analytical_lab_methods
The amount of chloride in an unknown sample is determined by potentiometric titration using two silver/silver-ion half-cells connected by a salt bridge. Half-cell A contains 1.0 M $Ag^+$ and half-cell B contains the analyte. A saline solution consisting of $NaCl$ in 5% dextrose is analyzed: 100.0 g of solution is place...
Potentiometric titration curve of cell voltage versus volume of 0.200 M AgNO3 added, including the sharp equivalence-region change and the voltage at 10.00 mL.
Calculate the mass percentage of $NaCl$ in the saline solution.
The endpoint of the titration takes place at $18.8 \text{ mL}$ added $\text{AgNO}_3$, so $$\text{mol } \text{Cl}^- = (0.0188 \text{ L})(0.200 \text{ mol L}^{-1}) = 3.76 \times 10^{-3} \text{ mol}$$ $$(3.76 \times 10^{-3} \text{ mol NaCl})(58.44 \text{ g mol}^{-1}) = 0.220 \text{ g NaCl}$$ $$(0.220 \text{ g NaCl})/(100...
train
calculation_with_work
context_first_then_domain_knowledge
505
olympiad
final
chemistry
analytical_lab_methods
The amount of chloride in an unknown sample is determined by potentiometric titration using two silver/silver-ion half-cells connected by a salt bridge. Half-cell A contains 1.0 M $Ag^+$ and half-cell B contains the analyte. A saline solution consisting of $NaCl$ in 5% dextrose is analyzed: 100.0 g of solution is place...
Potentiometric titration curve of cell voltage versus volume of 0.200 M AgNO3 added, including the sharp equivalence-region change and the voltage at 10.00 mL.
Calculate the concentration of free silver ion in half-cell B when 10.00 mL of titrant has been added.
At $10.00 \text{ mL}$ added titrant, the absolute value of the cell potential is $0.470 \text{ V}$. From the Nernst equation: $$E = E^\circ - (0.0591/n)\log([\text{Ag}^+]_{\text{sample}}/[\text{Ag}^+]_{\text{ref}})$$ $$0.470 \text{ V} = 0 \text{ V} - 0.0591 \cdot \log([\text{Ag}^+]_{\text{sample}}/[1])$$ $$[\text{Ag}^...
train
calculation_with_work
context_first_then_domain_knowledge
506
olympiad
final
chemistry
redox
The amount of chloride in an unknown sample is determined by potentiometric titration using two silver/silver-ion half-cells connected by a salt bridge. Half-cell A contains 1.0 M $Ag^+$ and half-cell B contains the analyte. A saline solution consisting of $NaCl$ in 5% dextrose is analyzed: 100.0 g of solution is place...
Potentiometric titration curve of cell voltage versus volume of 0.200 M AgNO3 added, including the sharp equivalence-region change and the voltage at 10.00 mL.
Calculate the $K_{sp}$ of $AgCl(s)$.
At the point where 10.00 mL AgNO<sub>3</sub> has been added, essentially all the added silver ion ($0.01000 \text{ L} \times 0.200 \text{ mol L}^{-1} = 2.00 \times 10^{-3} \text{ mol}$) has reacted to form AgCl(s) (since, according to part c, only about $10^{-9} \text{ mol Ag}^+$ is still present in solution). Thus the...
train
calculation_with_work
context_first_then_domain_knowledge
507
olympiad
final
chemistry
stoichiometry
The binary compounds $LiX$ ($X=H, F, Cl, Br,$ or $I$) crystallize in the rock-salt ($NaCl$) cubic lattice. The unit-cell edge length is denoted $a$; small circles represent Li and large circles represent X.
Rock-salt unit cell used to determine the number of LiCl formula units per cell.
Calculate the density of $\text{LiCl}$, $a = 513.6$ pm.
Volume of one unit cell = $a^3 = (513.6 \text{ pm})^3 = (513.6 \times 10^{-10} \text{ cm})^3 = 1.355 \times 10^{-22} \text{ cm}^3$ Mass of one unit cell = $4 \text{ LiCl} = 4(42.39 \text{ g mol}^{-1})/(6.022 \times 10^{23} \text{ mol}^{-1}) = 2.816 \times 10^{-22} \text{ g}$ Density = $(2.816 \times 10^{-22} \text{ g...
train
calculation_with_work
context_first_then_domain_knowledge
508
olympiad
final
chemistry
equilibrium
The $K_{sp}$ of $Ca(IO_3)_2$ was measured at several temperatures. The provided van't Hoff plot shows $\ln(K_{sp})$ as a function of $1/T$. For 317–363 K, the fitted line is $y=-330x-10.63$; for 273–317 K, the fitted line is $y=-11400x+24.3$.
A plot of ln(Ksp) of Ca(IO3)2 versus 1/T, K^-1. The x-axis ranges from 0.0026 to 0.0038. The y-axis ranges from -11 to -18. The curve shows a break. The first segment, corresponding to temperatures T = 317 - 363 K, has the linear equation y = -330x - 10.63. The second segment, corresponding to temperatures T = 273 - 31...
Calculate $\Delta H^\circ$ and $\Delta S^\circ$ for the dissolution of $Ca(IO_3)_2$ in the range of $T = 273$ to $317$ K.
The figure given is a van't Hoff plot for the dissolution reaction of $Ca(IO_3)_2$, since $K_{sp} = K_{eq}$ for the dissolution reaction. Since $\ln(K_{eq}) = -(\Delta G^\circ/RT) = (-\Delta H^\circ/R)(1/T) + (\Delta S^\circ/R)$, the slope of the plot is $-\Delta H^\circ/R$ and the intercept is $\Delta S^\circ/R$. $$\...
train
calculation_with_work
context_first_then_domain_knowledge
509
olympiad
final
chemistry
equilibrium
The $K_{sp}$ of $Ca(IO_3)_2$ was measured at several temperatures. The provided van't Hoff plot shows $\ln(K_{sp})$ as a function of $1/T$. For 317–363 K, the fitted line is $y=-330x-10.63$; for 273–317 K, the fitted line is $y=-11400x+24.3$.
A plot of ln(Ksp) of Ca(IO3)2 versus 1/T, K^-1. The x-axis ranges from 0.0026 to 0.0038. The y-axis ranges from -11 to -18. The curve shows a break. The first segment, corresponding to temperatures T = 317 - 363 K, has the linear equation y = -330x - 10.63. The second segment, corresponding to temperatures T = 273 - 31...
The $\ln(K_{sp})$ vs. $1/T$ plot has a break in the curve because the stable crystalline form of calcium iodate changes in the temperature range shown. At some temperatures, the stable form is anhydrous $Ca(IO_3)_2$, while at other temperatures the hexahydrate $Ca(IO_3)_2 \cdot 6 H_2O$ is the stable form. Assign which ...
Dissolution of calcium iodate in the 317-363 K range takes place with a much smaller $\Delta H^\circ$ ($2.7 \text{ kJ mol}^{-1}$) and much smaller (more negative) $\Delta S^\circ$ ($-88.38 \text{ J mol}^{-1} \text{ K}^{-1}$) than in the 273-317 K range discussed in (c). The hexahydrate will certainly dissolve with a mo...
train
calculation_with_work
context_first_then_domain_knowledge
510
olympiad
final
chemistry
redox
Ascorbic acid (Vitamin C, HAsc; $C_6H_8O_6$) is a weak acid with $pK_a=4.10$ and is readily oxidized to dehydroascorbic acid ($C_6H_6O_6$). Ferricyanide, $Fe(CN)_6^{3-}$, can carry out this oxidation. The reaction rate is followed by measuring $[Fe(CN)_6^{3-}]$ through its absorbance at 416 nm. Two runs use $[H^+]=0.0...
Graph showing ln([Fe(CN)_6^{3-}]) versus time (t, s). The x-axis ranges from 0 to 600 s. The y-axis ranges from -7.5 to -10. Two linear plots are shown. The upper line corresponds to [HAsc] = 0.00507 M, with the equation y = -7.56 - 0.00443 x. The lower line corresponds to [HAsc] = 0.0152 M, with the equation y = -7.62...
What is the rate constant for this reaction at $[H^+] = 0.06$ M? Make sure to include units.
$\text{Rate} = k[HAsc][Fe(CN)_6^{3-}] = k_{\text{obs}}[Fe(CN)_6^{3-}]$, where $k_{\text{obs}} = -\text{slope of the } \ln([Fe(CN)_6^{3-}]) \text{ vs. time plot}$. So $k[HAsc] = k_{\text{obs}}$, $k = k_{\text{obs}}/[HAsc]$. From the first run, $k = (0.00443 \text{ s}^{-1})/(0.00507 \text{ M}) = 0.874 \text{ M}^{-1} \tex...
train
brief_explanation
context_first_then_domain_knowledge
511
olympiad
final
chemistry
atomic_structure
The chemistry of beryllium (Be) differs substantially from that of its heavier Group 2 congener, barium (Ba). Explain each difference using atomic structure and bonding.
Crystal structures of solid $BeCl_2$ (left) and solid $BaCl_2$ (right); metal atoms are black and chlorine atoms are gray.
Solid $BeCl_2$ adopts the structure shown on the left, while solid $BaCl_2$ adopts the structure shown on the right.
Be can only form 4 bonds, so it can only adopt the left structure. The larger $\text{Ba}^{2+}$ ion can easily achieve the coordination number of 8 required by the right-hand structure.
train
brief_explanation
context_first_then_domain_knowledge
514
olympiad
final
chemistry
acids_bases
A sample of solid calcium fluoride is suspended in water in an unreactive container and stirred until equilibrium. The pH is lowered by careful addition of nitric acid. The graph shows pH and $[\mathrm{Ca}^{2+}]$; the concentration axis is in millimoles per liter.
Graph of $[\mathrm{Ca}^{2+}]$ in mmol L$^{-1}$ versus pH for a saturated $\mathrm{CaF_2}$ solution.
Determine the K_sp of CaF₂ from the data provided.
From the graph, the molar solubility of CaF₂ at high pH is about 0.21 mmol/L, so [Ca²⁺] = 2.1 × 10⁻⁴ M and [F⁻] = 2[Ca²⁺] = 4.2 × 10⁻⁴ M. Therefore, Ksp = [Ca²⁺][F⁻]² = (2.1 × 10⁻⁴)(4.2 × 10⁻⁴)² = 3.7 × 10⁻¹¹.
train
brief_explanation
context_first_then_domain_knowledge
515
olympiad
final
chemistry
acids_bases
A sample of solid calcium fluoride is suspended in water in an unreactive container and stirred until equilibrium. The pH is lowered by careful addition of nitric acid. The graph shows pH and $[\mathrm{Ca}^{2+}]$; the concentration axis is in millimoles per liter.
Graph of $[\mathrm{Ca}^{2+}]$ in mmol L$^{-1}$ versus pH for a saturated $\mathrm{CaF_2}$ solution.
Qualitatively, what is the cause for the increase in solubility of CaF₂ at low pH?
F⁻ reacts with H⁺ to form HF, reducing the concentration of F⁻ and shifting the solubility equilibrium so that more solid CaF₂ dissolves.
train
calculation_with_work
context_first_then_domain_knowledge
516
olympiad
final
chemistry
acids_bases
A sample of solid calcium fluoride is suspended in water in an unreactive container and stirred until equilibrium. The pH is lowered by careful addition of nitric acid. The graph shows pH and $[\mathrm{Ca}^{2+}]$; the concentration axis is in millimoles per liter.
Graph of $[\mathrm{Ca}^{2+}]$ in mmol L$^{-1}$ versus pH for a saturated $\mathrm{CaF_2}$ solution.
From the data provided, determine the K_a of HF.
At pH = 2, [Ca²⁺] = 1.36 × 10⁻³ M and [H₃O⁺] = 0.010 M. From Ksp, [F⁻] = 1.65 × 10⁻⁴ M, and [HF] = 2.56 × 10⁻³ M. Therefore, Ka = [H₃O⁺][F⁻]/[HF] = (0.010)(1.65 × 10⁻⁴)/(2.56 × 10⁻³) = 6.4 × 10⁻⁴.
train
calculation_with_work
context_first_then_domain_knowledge
517
olympiad
final
chemistry
acids_bases
A sample of solid calcium fluoride is suspended in water in an unreactive container and stirred until equilibrium. The pH is lowered by careful addition of nitric acid. The graph shows pH and $[\mathrm{Ca}^{2+}]$; the concentration axis is in millimoles per liter.
Graph of $[\mathrm{Ca}^{2+}]$ in mmol L$^{-1}$ versus pH for a saturated $\mathrm{CaF_2}$ solution.
How many moles of HNO₃ must be added to the CaF₂/water mixture to achieve a pH = 3.00 in this experiment? The volume of solution is 1.00 L.
At pH = 3, [Ca²⁺] = 3.9 × 10⁻⁴ M and [F⁻] + [HF] = 7.8 × 10⁻⁴ M. Also, [H₃O⁺] = 1.0 × 10⁻³ M. From Ka, [F⁻]/[HF] = 0.64. Solving gives [HF] = 4.8 × 10⁻⁴ M. Therefore, the moles of HNO₃ added are 4.8 × 10⁻⁴ + 1.0 × 10⁻³ = 1.5 × 10⁻³ mol.
train
calculation_with_work
context_first_then_domain_knowledge
518
olympiad
final
chemistry
acids_bases
A sample of solid calcium fluoride is suspended in water in an unreactive container and stirred until equilibrium. The pH is lowered by careful addition of nitric acid. The graph shows pH and $[\mathrm{Ca}^{2+}]$; the concentration axis is in millimoles per liter. Carbon dioxide dissolves in water at 25 °C and 1 atm to...
Graph of $[\mathrm{Ca}^{2+}]$ in mmol L$^{-1}$ versus pH for a saturated $\mathrm{CaF_2}$ solution.
What is the pH of the solution at this point? The K_sp of CaCO₃ is 8.7 × 10⁻⁹, the K_a of aqueous CO₂ ("H₂CO₃") is 4.3 × 10⁻⁷, and the K_a of HCO₃⁻ is 4.7 × 10⁻¹¹.
[Ca²⁺] = 2.1 × 10⁻⁴ M. From Ksp, [CO₃²⁻] = 4.1 × 10⁻⁵ M. Using the combined acid–base dissociation equilibria, [H₃O⁺] = 1.3 × 10⁻⁷ M, giving a pH of 6.88.
train
calculation_with_work
context_first_then_domain_knowledge
519
olympiad
final
chemistry
kinetics
Ethene, $C_2H_4$, reacts in the gas phase in the presence of radicals $R$ to extend a polyethylene radical chain. The forward reaction is second order and the reverse reaction is first order; both rate constants are independent of the degree of polymerization and the identity of $R$. The graph gives linear plots of $\l...
Arrhenius-style graph of $\ln(k)$ versus $1/T$ for $k_f$ and $k_r$, including the fitted line equations.
Calculate ΔH° and ΔS° for the polymerization reaction.
From the Arrhenius plots, ΔH° = −10,500 × 8.314 J/mol = −87.3 kJ/mol, and ΔS° = −16.5 × 8.314 J/mol·K = −137 J/mol·K.
train
calculation_with_work
context_first_then_domain_knowledge
520
olympiad
final
chemistry
kinetics
Ethene, $C_2H_4$, reacts in the gas phase in the presence of radicals $R$ to extend a polyethylene radical chain. The forward reaction is second order and the reverse reaction is first order; both rate constants are independent of the degree of polymerization and the identity of $R$. The graph gives linear plots of $\l...
Arrhenius-style graph of $\ln(k)$ versus $1/T$ for $k_f$ and $k_r$, including the fitted line equations.
What is the BDE of the carbon-carbon double bond in ethene?
ΔH° = BDE(C=C) − 2 × BDE(C−C) = −87.3 kJ/mol. BDE(C=C) = 2 × 345 kJ/mol − 87.3 kJ/mol = 603 kJ/mol.
train
numerical_value_with_units
context_first_then_domain_knowledge
521
olympiad
final
chemistry
kinetics
Ethene, $C_2H_4$, reacts in the gas phase in the presence of radicals $R$ to extend a polyethylene radical chain. The forward reaction is second order and the reverse reaction is first order; both rate constants are independent of the degree of polymerization and the identity of $R$. The graph gives linear plots of $\l...
Arrhenius-style graph of $\ln(k)$ versus $1/T$ for $k_f$ and $k_r$, including the fitted line equations.
What is the percent conversion of ethene into polymer at equilibrium under these conditions?
K_p = k_f/k_r = 0.147 at 720 K. At equilibrium, P_C2H4 = 6.8 bar. The percent conversion is (25 − 6.8)/25 × 100% = 73%.
train
equation_or_expression
context_first_then_domain_knowledge
522
olympiad
final
chemistry
kinetics
Ethene, $C_2H_4$, reacts in the gas phase in the presence of radicals $R$ to extend a polyethylene radical chain. The forward reaction is second order and the reverse reaction is first order; both rate constants are independent of the degree of polymerization and the identity of $R$. The graph gives linear plots of $\l...
Arrhenius-style graph of $\ln(k)$ versus $1/T$ for $k_f$ and $k_r$, including the fitted line equations.
By what factor does the catalyst accelerate the rate of the forward reaction at 500 K?
\[ k_f(\text{uncatalyzed},\ 500\ \mathrm{K}) = 1310\ \mathrm{bar^{-1}\,s^{-1}}. \quad k_f(\text{catalyzed},\ 500\ \mathrm{K}) = 2.96 \times 10^{6}\ \mathrm{bar^{-1}\,s^{-1}}. \quad \text{The catalyst accelerates the rate by a factor of } 2300. \]
train
equation_or_expression
context_first_then_domain_knowledge
523
olympiad
final
chemistry
kinetics
Ethene, $C_2H_4$, reacts in the gas phase in the presence of radicals $R$ to extend a polyethylene radical chain. The forward reaction is second order and the reverse reaction is first order; both rate constants are independent of the degree of polymerization and the identity of $R$. The graph gives linear plots of $\l...
Arrhenius-style graph of $\ln(k)$ versus $1/T$ for $k_f$ and $k_r$, including the fitted line equations.
By what factor does the catalyst change the rate of the reverse reaction at 500 K?
Since the catalyst does not change K_eq, the reverse reaction is also accelerated by a factor of 2300.
train
equation_or_expression
context_first_then_domain_knowledge
524
olympiad
final
chemistry
redox
Copper(II) forms $\mathrm{Cu(NH_3)_4^{2+}}$ with $K_f=1.7\times10^{13}$. At 298 K, half-cell A contains 100 mL of 1.00 M $\mathrm{Cu(NO_3)_2}$. Half-cell B contains 100 mL of a solution with a small amount of copper(II) and 0.100 M $\mathrm{NH_3}$. Nitric acid is added slowly to half-cell B while the cell potential is ...
Electrochemical concentration-cell diagram and measured cell potential versus volume of $\mathrm{HNO_3}$ added.
Which half-cell is the cathode and which is the anode? Justify your answer.
Half-cell B has a much lower concentration of \(\mathrm{Cu^{2+}(aq)}\). Since current will flow spontaneously in a concentration cell in a direction that tends to equalize the concentrations in the two half-cells, reduction of \(\mathrm{Cu(II)}\) will happen preferentially in half-cell A, which is thus the cathode. Hal...
train
brief_explanation
context_first_then_domain_knowledge
525
olympiad
final
chemistry
redox
Copper(II) forms $\mathrm{Cu(NH_3)_4^{2+}}$ with $K_f=1.7\times10^{13}$. At 298 K, half-cell A contains 100 mL of 1.00 M $\mathrm{Cu(NO_3)_2}$. Half-cell B contains 100 mL of a solution with a small amount of copper(II) and 0.100 M $\mathrm{NH_3}$. Nitric acid is added slowly to half-cell B while the cell potential is ...
Electrochemical concentration-cell diagram and measured cell potential versus volume of $\mathrm{HNO_3}$ added.
Qualitatively explain the shape of the graph.
As \(\mathrm{HNO_3}\) is added, it protonates \(\mathrm{NH_3}\) and decreases its concentration. Less \(\mathrm{NH_3}\) means a higher concentration of uncomplexed \(\mathrm{Cu^{2+}}\) and hence a smaller voltage. When all the \(\mathrm{NH_3}\) is protonated, then all the \(\mathrm{Cu(II)}\) is in the form of \(\mathrm...
train
brief_explanation
context_first_then_domain_knowledge
533
psat
standalone
maths
Geometry
In the figure, RT = TU, the measure of angle VST is 29°, and the measure of angle RVS is 41°. What is the value of x?
null
What is the value of x?
156
train
numerical_value
context_first_then_domain_knowledge
534
psat
standalone
maths
Statistics and Probability
The scatterplot shows the relationship between two variables, x and y. A line of best fit is also shown. For how many of the 11 data points does the line of best fit predict a greater y - value than the actual y - value?
null
For how many of the 11 data points does the line of best fit predict a greater y - value than the actual y - value?
6
train
numerical_value
context_first_then_domain_knowledge
535
psat
standalone
maths
Algebra
The parabola shown intersects the y - axis at the point (x, y). What is the value of y?
null
What is the value of y?
7
train
numerical_value
context_first_then_domain_knowledge
536
psat
standalone
maths
Algebra
Line j is shown in the xy - plane. Line k (not shown) is parallel to line j. What is the slope of line k?
null
What is the slope of line k?
4
train
numerical_value
context_first_then_domain_knowledge
539
olympiad
final
physics
electricity_magnetism
Two infinitely long concentric hollow cylinders have radii $a$ and $4a$ . Both cylinders are insulators; the inner cylinder has a uniformly distributed charge per length of $+\lambda$ ; the outer cylinder has a uniformly distributed charge per length of $-\lambda$ . An infinitely long dielectric cylinder with per...
The diagram shows two concentric infinitely long hollow cylinders with radii a and 4a. The inner cylinder is connected to a source and has a shaded annulus representing the dielectric region.
Determine the magnetic field for all regions.
We can apply the results of the previous section to obtain the enclosed charge density $\lambda_{in}$ as a function of radius, $$\lambda_{in} = \begin{cases} 0 & r < a \\ \lambda & a < r < 2a \\ \lambda/\kappa & 2a < r < 3a \\ \lambda & 3 < r < 4a \\ 0 & r > 4a. \end{cases}$$ Defining $$\lambda_i = \left(1 - \frac...
train
calculation_with_work
context_first_then_domain_knowledge
540
olympiad
final
physics
mechanics
Two masses $m$ separated by a distance $l$ are given initial velocities $v_0$ as shown in the diagram. The masses interact only through universal gravitation.
The diagram shows two masses, m, separated by a distance l. The mass on the left has an initial velocity v_0 pointing upwards. The mass on the right has an initial velocity v_0 pointing downwards.
Under what conditions will the masses eventually collide?
Intuitively, it is impossible for the masses to collide unless $v_0 = 0$ , by angular momentum conservation. It's okay to simply write this down, but it's worth taking a closer look, as this intuitive argument is *not* true for all central forces. By angular momentum conservation, $vr$ is constant where $r$ is th...
train
calculation_with_work
context_first_then_domain_knowledge
541
olympiad
final
physics
mechanics
Two masses $m$ separated by a distance $l$ are given initial velocities $v_0$ as shown in the diagram. The masses interact only through universal gravitation.
The diagram shows two masses, m, separated by a distance l. The mass on the left has an initial velocity v_0 pointing upwards. The mass on the right has an initial velocity v_0 pointing downwards.
Under what conditions will the masses follow circular orbits of diameter $l$ ?
In this case, the masses undergo uniform circular motion with radius $l/2$ and speed $v_0$ , so that $$\frac{Gm^2}{l^2} = \frac{mv_0^2}{l/2} \Rightarrow \frac{Gm}{v_0^2 l} = 2.$$
train
calculation_with_work
context_first_then_domain_knowledge
542
olympiad
final
physics
mechanics
Two masses $m$ separated by a distance $l$ are given initial velocities $v_0$ as shown in the diagram. The masses interact only through universal gravitation.
The diagram shows two masses, m, separated by a distance l. The mass on the left has an initial velocity v_0 pointing upwards. The mass on the right has an initial velocity v_0 pointing downwards.
Under what conditions will the masses follow closed orbits?
The masses follow closed orbits if they do not have enough energy to escape to infinity, i.e. if the total energy of the system is negative, $$E = 2 \cdot \frac{1}{2}mv_0^2 - \frac{Gm^2}{l} < 0 \Rightarrow \frac{Gm}{v_0^2 l} > 1.$$
train
calculation_with_work
context_first_then_domain_knowledge
543
olympiad
final
physics
mechanics
Two masses $m$ separated by a distance $l$ are given initial velocities $v_0$ as shown in the diagram. The masses interact only through universal gravitation.
The diagram shows two masses, m, separated by a distance l. The mass on the left has an initial velocity v_0 pointing upwards. The mass on the right has an initial velocity v_0 pointing downwards.
What is the minimum distance achieved between the masses along their path?
Note that the masses will always move symmetrically about the center of mass. Thus, in order to be at minimum separation, their velocities must be perpendicular to the line joining them (and will be oppositely directed). Let the minimum separation be $d$ , and let the speed of each mass at minimum separation be $v$ ....
train
calculation_with_work
context_first_then_domain_knowledge
544
olympiad
final
physics
thermal_physics
The flow of heat through a material can be described via the *thermal conductivity* $\kappa$ . If the two faces of a slab of material with thermal conductivity $\kappa$ , area $A$ , and thickness $d$ are held at temperatures differing by $\Delta T$ , the thermal power $P$ transferred through the slab is $$P = ...
Diagram of a cross-section of a heat exchanger. It shows two parallel tubes of length l, width w, and height h. The tubes are separated by a metal wall of thickness d. Hot fluid (red) flows down the lower tube at speed v. Cold fluid (blue) flows up the upper tube at speed v.
Find $\Delta T_f$ in terms of the other given parameters.
To see why the temperature difference across the wall is the same everywhere along the wall, note that at every point along the wall, the warmer fluid on one side transfers energy to the colder fluid on the other. Since the heat capacities are equal, the temperature of the warmer fluid drops at the same rate the temper...
train
calculation_with_work
context_first_then_domain_knowledge
545
olympiad
final
physics
mechanics
A graduated cylinder is partially filled with water; a rubber duck floats at the surface. Oil is poured into the graduated cylinder at a slow, constant rate, and the volume marks corresponding to the surface of the water and the surface of the oil are recorded as a function of time. Water has a density of 1.00 g/mL; ...
Graph showing Volume (mL) versus Time (min). The x-axis (Time) ranges from 0 to 10 minutes. The y-axis (Volume) ranges from 0 to 500 mL. The data points show two distinct trends: 'Volume mark at water surface' (marked with 'x') decreases linearly from approximately 250 mL at t=0 to 100 mL at t=10 min. 'Volume mark at o...
Find the density of the oil.
As the oil is poured in, more and more of the weight of the duck is supported by oil, and it rises out of the water, reducing the water level. Eventually this stops, either because the duck is fully submerged in oil or because it is floating entirely above the water; there is not sufficient information to tell which. A...
train
calculation_with_work
context_first_then_domain_knowledge
546
olympiad
final
physics
mechanics
Shown below is the *Blackbird*, a vehicle built in 2009. There is no source of stored energy such as a battery or gasoline engine; all of the power used to move the car comes from the wind. The only important mechanism in the car is a gearbox that can transfer power between the wheels and the propeller. The *Blackbir...
Photo of the Blackbird vehicle, a car with a large vertical propeller mounted on the back, driving on a road.
Consider first the **downwind faster than the wind** case. - Is the motion actually possible as claimed? If not, offer a brief explanation! - If the motion is possible, is power transferred from the propeller to the wheels or vice versa? - If the motion is possible, what ground speed is attained? For this question, su...
**Both modes are possible as claimed.** The solution uses the idea of “center-of-mass (CoM) power,” which we will review below for interested readers; the solution itself starts on the next page. For simplicity, we will work in one dimension. If a particle moves with speed (v_1) and experiences a force (F), then the r...
train
calculation_with_work
context_first_then_domain_knowledge
547
olympiad
final
physics
thermal_physics
A room air conditioner is modeled as a heat engine run in reverse: an amount of heat $Q_L$ is absorbed from the room at a temperature $T_L$ into cooling coils containing a working gas; this gas is compressed adiabatically to a temperature $T_H$ ; the gas is compressed isothermally in a coil outside the house, givi...
Diagram of the reverse heat-engine air-conditioner cycle, showing the room cooling coil, pump, outside heating coil, and valve.
Derive an expression for the maximum rate at which heat is removed from the room in terms of the air temperatures $T_H$ , $T_L$ , and the power consumed by the air conditioner $P$ . Your derivation must refer to the entropy changes that occur in a Carnot cycle in order to receive full marks for this part.
The optimal performance is attained by a Carnot cycle running in reverse. Since a Carnot cycle is reversible, it keeps the total entropy of the heat reservoirs constant. The change in entropy for a reservoir of temperature $T$ absorbing heat $Q$ is $\Delta S = Q/T$ , so $$\frac{Q_H}{T_H} = \frac{Q_L}{T_L}.$$ Ene...
train
calculation_with_work
context_first_then_domain_knowledge
548
olympiad
final
physics
electricity_magnetism
A positive point charge $q$ is located inside a neutral hollow spherical conducting shell. The shell has inner radius $a$ and outer radius $b$ ; $b - a$ is not negligible. The shell is centered on the origin. The point charge $q$ is located at the origin in the very center of the shell.
Cross-section of the concentric hollow spherical conducting shell, with inner radius $a$ and outer radius $b$ .
Determine the magnitude of the electric field outside the conducting shell at $x = b$ .
We apply Gauss's law for a sphere with radius $r > b$ centered about the origin. Since the shell is neutral, the enclosed charge is $q$ , so by spherical symmetry $$E(r) = \frac{q}{4\pi\epsilon_0 r^2}$$ outside the shell. Just outside the shell, the field is $q/(4\pi\epsilon_0 b^2)$ .
train
calculation_with_work
context_first_then_domain_knowledge
549
olympiad
final
physics
mechanics
A block of mass $M$ has a hole drilled through it so that a ball of mass $m$ can enter horizontally and then pass through the block and exit vertically upward. The ball and block are located on a frictionless surface; the block is originally at rest.
Diagram of the ball entering the block horizontally through the side hole and exiting vertically through the top hole.
Consider the scenario where the ball is traveling horizontally with a speed $v_0$ . The ball enters the block and is ejected out the top of the block. Assume there are no frictional losses as the ball passes through the block, and the ball rises to a height much higher than the dimensions of the block. The ball then r...
After the collision, the ball and block have the same horizontal velocity $v_1$ . Since the horizontal momentum is conserved, $$v_1 = \frac{m}{m+M}v_0.$$ Let $v_2$ be the vertical component of the velocity of the ball immediately after the collision. Since there are no frictional losses, conservation of energy yie...
train
calculation_with_work
context_first_then_domain_knowledge
550
olympiad
final
physics
electricity_magnetism
Consider the circuit shown below. $I_s$ is a constant current source, meaning that no matter what device is connected between points A and B, the current provided by the constant current source is the same.
Circuit with a constant current source $I_s$ feeding a resistor network whose output terminals are A and B; the branch resistances are labeled $2R$ and $4R$.
Connect an ideal voltmeter between A and B. Determine the voltage reading in terms of any or all of $R$ and $I_s$ .
An ideal voltmeter has infinite resistance, so no current flows between A and B. By symmetry, the same current must flow down each leg, so the current in each leg is $I_s/2$ . Assume the potential at the bottom is zero. The potential at A is the same as the junction to the left of A, so $$V_A = \frac{I_s}{2} 2R = I_...
train
calculation_with_work
context_first_then_domain_knowledge
551
olympiad
final
physics
electricity_magnetism
Consider the circuit shown below. $I_s$ is a constant current source, meaning that no matter what device is connected between points A and B, the current provided by the constant current source is the same.
Circuit with a constant current source $I_s$ feeding a resistor network whose output terminals are A and B; the branch resistances are labeled $2R$ and $4R$.
Connect instead an ideal ammeter between A and B. Determine the current in terms of any or all of $R$ and $I_s$ .
An ideal ammeter has zero resistance, so we just need to find the current through the effective $6R$ resistor that connects the two vertical branches. This current will flow to the left. By symmetry, the current through each vertical resistance of $2R$ must be the same, as well as the currents through each vertica...
train
calculation_with_work
context_first_then_domain_knowledge
552
olympiad
final
physics
electricity_magnetism
The original circuit is driven by a constant current source $I_s$ . After the source, the current splits into two resistor branches. Each branch contains resistors labeled $4R$ and $2R$ , and terminals A and B are the intermediate nodes between those resistors. The current supplied by $I_s$ is constant regardless...
Norton-equivalent circuit between terminals A and B: a current source $I_t$ in parallel with a resistor $R_t$.
It turns out that it is possible to replace the above circuit with a new circuit as follows: From the point of view of *any* passive resistance that is connected between A and B the circuits are identical. You don't need to prove this statement, but you do need to find $I_t$ and $R_t$ in terms of any or all of $R...
We can simply use the previous results. If A and B are shorted, all of the current will flow through AB, so $I_t = I_6 = I_s/9$ . If the resistance between A and B is infinite, the potential across AB will be $I_sR$ , so $R_t = 9R$ . The statement we made above is called Norton's theorem.
train
calculation_with_work
context_first_then_domain_knowledge
553
olympiad
final
physics
electricity_magnetism
A uniform magnetic cylinder has radius $R$ and length $L$ . It is composed of a very large number of densely packed microscopic dipoles, all aligned with the $z$ axis and pointing in the same direction. The dipoles are either all Ampère type or all Gilbert type, and the two models have the same external dipole fie...
Two microscopic dipole models in a cube of side $d$: a Gilbert model with negative and positive magnetic charges on opposite faces, and an Ampère model with oppositely directed surface currents on opposite faces.
Assume that $R \gg L$ and only Gilbert type dipoles, determine the magnitude and direction of $B$ at the center of the cylinder in terms of any or all of $p_m$ , $R$ , $L$ , $v_m$ , and any necessary fundamental constants.
The monopoles that make up the dipoles cancel out except on the flat surfaces. Then the cylinder acts like a parallel plate capacitor. If the size of a dipole is $d$ , then the surface density of monopole charge is $$\sigma_m = q_m/d^2.$$ Using the analogy with a parallel plate capacitor, the magnitude of $B$ is ...
train
calculation_with_work
context_first_then_domain_knowledge
554
olympiad
final
physics
electricity_magnetism
A uniform magnetic cylinder has radius $R$ and length $L$ . It is composed of a very large number of densely packed microscopic dipoles, all aligned with the $z$ axis and pointing in the same direction. The dipoles are either all Ampère type or all Gilbert type, and the two models have the same external dipole fie...
Two microscopic dipole models in a cube of side $d$: a Gilbert model with negative and positive magnetic charges on opposite faces, and an Ampère model with oppositely directed surface currents on opposite faces.
Assume that $R \ll L$ and only Ampère type dipoles, determine the magnitude and direction of $B$ at the center of the cylinder in terms of any or all of $p_m$ , $R$ , $L$ , $v_m$ , and any necessary fundamental constants.
The currents that make up the dipoles all cancel out except on the cylindrical surfaces. Then the cylinder acts like a solenoid, with $$B = \frac{\mu_0 I}{d}$$ where $I/d$ is the surface current density. The magnitude of $B$ is $$B = \frac{\mu_0 I}{d} = \mu_0 \frac{p_m}{d^3} = \mu_0 \frac{p_m}{v_m}.$$ The direc...
train
calculation_with_work
context_first_then_domain_knowledge
555
olympiad
final
physics
waves
The Doppler effect for a source moving relative to a stationary observer is described by $$f = \frac{f_0}{1 - (v/c) \cos \theta}$$ where $f$ is the frequency measured by the observer, $f_0$ is the frequency emitted by the source, $v$ is the speed of the source, $c$ is the wave speed, and $\theta$ is the angle between...
Graph of observed sound frequency versus time as a constant-frequency source moves past a stationary observer.
What is the speed of the source?
For $\theta = 0$ we have $$f_a \approx f_0/(1 - v/c)$$ and for $\theta = \pi$, $$f_b = f_0/(1 + v/c).$$ Read $f_a$ and $f_b$ from the early- and late-time portions of the graph and use $$\frac{f_a}{f_b} = \frac{1 + v/c}{1 - v/c},$$ giving $$v = 10.7\text{ m/s}.$$ Alternatively, since $v \ll c$, $$\frac{f_a}{f_...
train
calculation_with_work
context_first_then_domain_knowledge
556
olympiad
final
physics
waves
The Doppler effect for a source moving relative to a stationary observer is described by $$f = \frac{f_0}{1 - (v/c) \cos \theta}$$ where $f$ is the frequency measured by the observer, $f_0$ is the frequency emitted by the source, $v$ is the speed of the source, $c$ is the wave speed, and $\theta$ is the angle between...
Graph of observed sound frequency versus time as a constant-frequency source moves past a stationary observer.
What is the smallest distance between the source and the observer?
Let $d$ be the fixed distance between the observer and the source's path, and let $x$ be displacement along the path, with $x = 0$ at closest approach. For $|x| \ll d$, $$\cos \theta \approx \cot \theta = \frac{x}{d},$$ so $$f = \frac{f_0}{1 - (v/c)(x/d)} \approx f_0\left(1 + \frac{v}{c}\frac{x}{d}\right).$$ Taking...
train
calculation_with_work
context_first_then_domain_knowledge
557
olympiad
final
physics
electricity_magnetism
A simple integrated circuit device has two inputs, $V_a$ and $V_b$, and two outputs, $V_o$ and $V_g$. The inputs are effectively connected internally by an infinite resistance, so no current flows between them. The outputs are connected internally to a perfect source of emf $\mathcal{E}$, configured so that $$\mathcal...
Circuit for a non-inverting op-amp amplifier with feedback resistors $R_1$ and $R_2$.
Consider the following circuit. $R_1 = 8.2 \text{ k}\Omega$ and $R_2 = 560 \Omega$ are two resistors. Terminal $g$ and the negative side of $V_{in}$ are connected to ground, so both are at a potential of 0 volts. Determine the ratio $V_{out}/V_{in}$.
Since terminal $g$ is grounded, $V_g = 0$ and $V_a = V_{in}$, so $V_{out} = G(V_{in} - V_b)$. No current runs between $a$ and $b$, so any current through $R_1$ also flows through $R_2$. Ohm's law gives $$\frac{V_b}{R_2} = \frac{V_{out}}{R_1 + R_2} \Rightarrow V_{out} = G\left(V_{in} - V_{out}\frac{R_2}{R_1 + R_2}\righ...
train
calculation_with_work
context_first_then_domain_knowledge
558
olympiad
final
physics
electricity_magnetism
A simple integrated circuit device has two inputs, $V_a$ and $V_b$, and two outputs, $V_o$ and $V_g$. The inputs are effectively connected internally by an infinite resistance, so no current flows between them. The outputs are connected internally to a perfect source of emf $\mathcal{E}$, configured so that $$\mathcal...
Circuit for an op-amp subtractor using four equal resistors $R$ and input voltages $V_1$ and $V_2$.
Consider the circuit below, where all four resistors have resistance $R$. Note that the positions of terminals $a$ and $b$ have been switched. Determine $V_{out}$ in terms of $V_1$, $V_2$, and $R$.
Negative feedback drives $V_a \approx V_b$. Here the output is connected back to input $b$, so this is a stable equilibrium. Again $V_g = 0$. Since the voltage drops across the two lower resistors are equal, $$V_a = \frac{V_2}{2}.$$ Similarly, the voltage drops across the two upper resistors are equal, so $$V_1 + V...
train
calculation_with_work
context_first_then_domain_knowledge
559
olympiad
final
physics
electricity_magnetism
A simple integrated circuit device has two inputs, $V_a$ and $V_b$, and two outputs, $V_o$ and $V_g$. The inputs are effectively connected internally by an infinite resistance, so no current flows between them. The outputs are connected internally to a perfect source of emf $\mathcal{E}$, configured so that $$\mathcal...
Op-amp differentiator circuit containing an input capacitor $C$ and feedback resistor $R$.
Now consider a circuit with a capacitor $C$ and a resistor $R$ with time constant $RC = \tau$, where $V_{in}(t)$ smoothly varies in time. Determine $V_{out}$ in terms of $V_{in}(t)$ and $\tau$.
Negative feedback drives $V_b \approx V_a = 0$. The capacitor charge and current satisfy $$Q = CV_{in}, \qquad \dot{Q} = -\frac{V_{out}}{R},$$ where the second result follows from Ohm's law. Thus $$\frac{V_{out}}{R} = -C\frac{dV_{in}}{dt} \Rightarrow V_{out} = -\tau\frac{dV_{in}}{dt}.$$ This circuit is a differenti...
train
calculation_with_work
context_first_then_domain_knowledge
560
olympiad
final
physics
mechanics
null
Loop-the-loop track with a rolling ball released from height $h$, loop radius $R$, and landing point P at the bottom of the loop.
A uniform solid spherical ball starts from rest on a loop-the-loop track. It rolls without slipping along the track. However, it does not have enough speed to make it to the top of the loop. From what height $h$ would the ball need to start in order to land at point P directly underneath the top of the loop? Express yo...
Take the origin at P and let the ball leave the loop at angle $\theta$ from the vertical. Its coordinates are $$x = R\sin\theta, \qquad y = R(1+\cos\theta).$$ Energy conservation gives $$mg(h-y) = \frac12 mv^2 + \frac12 I\omega^2 = \frac12 m(1+\beta)v^2,$$ where $\beta = 2/5$ for a uniform solid sphere. At release...
train
calculation_with_work
context_first_then_domain_knowledge
561
olympiad
final
physics
mechanics
A pair of wedges are located on a horizontal surface. The coefficient of friction (both sliding and static) between the wedges is $\mu$ , the coefficient of friction between the bottom wedge B and the horizontal surface is $\mu$ , and the angle of the wedge is $\theta$ . The mass of the top wedge A is $m$ , and the...
Diagram showing two wedges A and B on a horizontal surface. Wedge A is on top of wedge B. A horizontal force F is applied to wedge B, directed to the left. The angle of the incline of wedge B is theta.
Determine the range of values for $F$ so that the top wedge does not slip on the bottom wedge. Express your answer(s) in terms of any or all of $m$ , $g$ , $\theta$ , and $\mu$ .
**Solution 1.** Assume the block does not slip. Considering the horizontal forces on the entire system gives $$F - 3\mu mg = 3ma \Rightarrow a = \frac{F}{3m} - \mu g.$$ When $F$ is small, the top wedge wants to slide downward, so static friction points up the ramp. Considering the horizontal and vertical forces on ...
train
calculation_with_work
context_first_then_domain_knowledge
562
olympiad
final
physics
electricity_magnetism
A pair of iron balls placed in the same horizontal plane as the compass but a distance $d$ away can be used to help correct for the error caused by the induced magnetism of the ship. Just like the ship, the iron balls will become magnetic because of the Earth's field $B_e$ . As spheres, the balls will individually ...
Diagram of the iron-ball dipole geometry, showing the north direction, dipole separation $a$, field point, radial direction, magnetic field $\vec{B}_i$, and angle $\phi$.
Derive an expression for the magnetic field $\vec{B}_i$ from the iron a distance $d \gg a$ from the center of the ball. Note that there will be a component directed radially away from the ball and a component directed tangent to a circle of radius $d$ around the ball, so using polar coordinates is recommended.
This problem is not nearly as difficult as it looks. Consider the colored triangle above. The black side has length $a$ . The angle between the green and black sides is $\phi$ , so the length of the red side is $a \sin \phi$ and the length of the green side is $a \cos \phi$ . The magnetic field strength from one...
train
calculation_with_work
context_first_then_domain_knowledge
563
olympiad
final
physics
mechanics
Suppose a domino stands upright on a table. It has height $h$ , thickness $t$ , width $w$ , and mass $m$ . The domino is free to rotate about its edges, but will not slide across the table. a. Suppose we give the domino a sharp, horizontal impulsive push with total momentum $p$ .
Diagram of the upright domino with dimensions $h$, $t$, and $w$, and a horizontal impulse $p$ applied at height $H$.
At what height $H$ above the table is the impulse $p$ required to topple the domino smallest?
First we'll look for the height $H$ at which we should push to topple the domino with the least momentum. A convenient method is to look at the angular momentum in the domino because this is easy to calculate and describes rotational motion. Because the push is horizontal, the moment arm of the push (about the domino...
train
calculation_with_work
context_first_then_domain_knowledge
564
olympiad
final
physics
mechanics
Suppose a domino stands upright on a table. It has height $h$ , thickness $t$ , width $w$ , and mass $m$ . The domino is free to rotate about its edges, but will not slide across the table. a. Suppose we give the domino a sharp, horizontal impulsive push with total momentum $p$ .
Diagram of the upright domino with dimensions $h$, $t$, and $w$, and a horizontal impulse $p$ applied at height $H$.
What is the minimum value of $p$ to topple the domino?
Next we calculate the minimum $p$ using energy. As the domino rotates, it converts kinetic energy to potential energy, so we'll calculate both. The domino's potential energy is greatest when its center of mass is directly over the contact point of the domino and the table. That height is half the diagonal of the domi...
train
calculation_with_work
context_first_then_domain_knowledge
565
olympiad
final
physics
physics_general
Beloit College's proton accelerator uses a tuning electromagnet made from spiral-wound copper pipe.
Cross-section of the flat spiral winding, labeled with inner diameter $D_i$ and outer diameter $D_o$.
The electromagnet is composed of layers of spiral wound copper pipe; the pipe has inner diameter $d_i = 0.40$ cm and outer diameter $d_o = 0.50$ cm. The copper pipe is wound into this flat spiral that has an inner diameter $D_i = 20$ cm and outer diameter $D_o = 50$ cm. Assuming the pipe almost touches in the s...
Treat the problem as two dimensional. The area of the spiral is $$A = \frac{\pi}{4}(D_o^2 - D_i^2).$$ The area of the pipe is $$A = L d_o.$$ Equating and solving, $$L = \frac{\pi(D_o^2 - D_i^2)}{4d_o} = 33 \text{ m.}$$
train
calculation_with_work
context_first_then_domain_knowledge
570
olympiad
final
physics
waves
A square room has side length $L$. Its bottom wall is a perfect mirror, so reflected light acquires a phase shift of $180^\circ$. A monochromatic point source of wavelength $\lambda$ is located a distance $d$ above the center of the mirror, with $\lambda\ll d\ll L$.
Square-room geometry with a point source above the center of a perfectly reflecting bottom wall.
On the right wall, an interference pattern emerges. What is the distance $y$ between the bottom corner and the closest bright fringe above it? You may assume $\lambda\ll y\ll L$.
The reflected wave can be treated as coming from an image source a distance $d$ below the mirror. For a point at height $y$ on the right wall, the path-length difference is $$\Delta\ell \approx\frac{4dy}{L}.$$ Reflection adds a phase shift equivalent to an additional path difference $\lambda/2$, so bright fringes sat...
train
calculation_with_work
context_first_then_domain_knowledge
571
olympiad
final
physics
experimental_physics
A square room has side length $L$. Its bottom wall is a perfect mirror, so reflected light acquires a phase shift of $180^\circ$. A monochromatic point source of wavelength $\lambda$ is located a distance $d$ above the center of the mirror, with $\lambda\ll d\ll L$. For the experiment, $L=40\text{ m}$ and $550\text{ n...
Square-room mirror geometry used to relate source height and fringe positions.
At what $d$ should the point source be placed to minimize the uncertainty in the measured wavelength? Roughly what is the minimum uncertainty?
The fractional uncertainty is $$\frac{\Delta\lambda}{\lambda} = \sqrt{ \left(\frac{\Delta d}{d}\right)^2+ \left(\frac{\Delta y_{10}}{y_{10}}\right)^2 }.$$ The tenth bright fringe has $$y_{10}=\frac{19\lambda L}{8d},$$ so $d\,y_{10}$ is fixed. Equal absolute ruler uncertainties are minimized when $$d=y_{10}.$$ Usi...
train
calculation_with_work
context_first_then_domain_knowledge
732
olympiad
final
earth_science
geosphere
The Western Alps features a zone of intense deformation and offers a rare example of exhumed ultra-high pressure (UHP) metamorphic rock. Shown in Figure 1 is a geologic map of a small section of the Western Alps, with rock units labelled A through E. Letter x denotes a region of folding and faulting, of which a small s...
null
Using Figure 2, classify fault F as either normal, reverse, or thrust. Give your answer as N, R, or T and assume no overturning of strata.
R
train
short_answer
context_first_then_domain_knowledge
733
olympiad
final
earth_science
atmosphere
The energy driving surface processes and life on Earth is almost entirely derived from incoming solar radiation, or insolation. The balance of energy entering and exiting, sometimes referred to as Earth's heat budget, is highly relevant in climate studies.
Figure 1: Global map showing Top of Atmosphere (TOA) insolation (Solar Radiation Budget, SRB) as monthly average SW Downward Flux (W m⁻²) from 1983 to 2007. The map uses a color scale ranging from 160 W/m² (dark blue/purple) at high latitudes to 400 W/m² (red/orange) at the equator. Insolation decreases towards the pol...
Give a brief reason for the following: The overall insolation is lower at the surface compared to the TOA.
Incoming solar radiation must pass through the atmosphere, where it is at least partially scattered, reflected, and absorbed, before reaching the surface. Therefore, surface insolation will be less than the insolation incident on the top of the atmosphere.
train
brief_explanation
context_first_then_domain_knowledge
734
olympiad
final
earth_science
atmosphere
The energy driving surface processes and life on Earth is almost entirely derived from incoming solar radiation, or insolation. The balance of energy entering and exiting, sometimes referred to as Earth's heat budget, is highly relevant in climate studies.
Figure 1: Global map showing Top of Atmosphere (TOA) insolation (Solar Radiation Budget, SRB) as monthly average SW Downward Flux (W m⁻²) from 1983 to 2007. The map uses a color scale ranging from 160 W/m² (dark blue/purple) at high latitudes to 400 W/m² (red/orange) at the equator. Insolation decreases towards the pol...
Give a brief reason for the following: The difference between TOA and surface insolation is non-uniform.
Circulation patterns in the atmosphere and ocean influence the degree of absorption locally.
train
brief_explanation
context_first_then_domain_knowledge
735
olympiad
final
earth_science
atmosphere
The energy driving surface processes and life on Earth is almost entirely derived from incoming solar radiation, or insolation. The balance of energy entering and exiting, sometimes referred to as Earth's heat budget, is highly relevant in climate studies.
Figure 2: Global map showing Surface insolation (Solar Radiation Budget, SRB) in W/m². The map highlights high insolation values (yellow/orange) across the equatorial region and mid-latitudes, and lower values (blue/cyan) near the poles. An arrow labeled 'A' points towards the equatorial region, indicating high insolat...
Based on your knowledge of the atmosphere, provide one reason for the long region of relatively low insolation at Location A.
Location A coincides with the intertropical convergence zone, where the lower limb of Hadley circulation rises near the equator and forms a band of frequent thunderstorms. These dense clouds reflect light, preventing much of it from reaching the surface.
train
brief_explanation
context_first_then_domain_knowledge
736
olympiad
final
earth_science
atmosphere
The role of clouds in climate change is one of the most complex and uncertain in climate modeling. For instance, different types of clouds differ in interaction with radiation.
Figure 3: Spectra of radiation from the Sun and Earth (top), absorption spectra of a few atmospheric components (water vapor, carbon dioxide, etc.) (mid), and total absorption spectrum of the atmosphere (bottom).
Cirrus clouds tend to have a warming effect as a result of the absorption spectrum of ice. With this information, give an approximate range of wavelengths (in micrometers) for the atmospheric window that cirrus clouds act to partially close.
About 8-12 micrometers. Cirrus clouds, composed primarily of ice crystals, absorb longwave radiation that would escape to space in their absence.
train
brief_explanation
context_first_then_domain_knowledge
End of preview. Expand in Data Studio

HighSchoolBench-OEQ

GitHub • Master’s thesis

Dataset description

HighSchoolBench-OEQ contains 2,952 open-ended questions from high-school standardized assessments and science competitions. It is the open-ended component of HighSchoolBench, a multimodal benchmark for evaluating language and vision-language models on high-school assessment questions.

The questions come from assessments such as AP, IB, SAT, PSAT, NAEP, and PISA, U.S. state examinations, and U.S. science olympiads. They include text-based questions and questions involving images, diagrams, and tables.

Each question includes a free-text reference answer and metadata identifying its source assessment, subject, domain, and expected answer type. Answers range from numerical values and mathematical expressions to explanations and worked solutions. The multiple-choice component is available separately as HighSchoolBench-MCQ.

Dataset statistics

The dataset contains 2,952 questions and 531 images.

Split Questions
train 449
validation 141
test 2,362
Total 2,952

The source split val is exposed as validation on Hugging Face. The split field retains the original source label.

Configurations

The dataset is organized into five configurations according to the supporting context. All configurations contain free-text reference answers.

Configuration Supporting context
context_text_only__text_response Text
context_image_only__text_response Images
context_table_only__text_response Tables
context_table_image__text_response Tables and images
context_text_contextualized_image__text_response Images with associated passages

The context category describes the supporting material accompanying the question; the question stem is stored separately in question.

In the contextualized configuration, individual images are paired with their associated passages. This configuration contains train and test splits; the other configurations also include validation.

Data format

Main fields

Field Description
id Question identifier from the source file.
assessment_name Source assessment or competition.
assessment_type Assessment stage or exam type.
subject Subject assigned to the question.
domain Topic within the subject.
context Supporting passage or shared context, when available.
question Question stem. Mathematical expressions use LaTeX.
answer Free-text reference answer.
split Original split label: train, val, or test.
answer_type Expected form of the answer.
context_policy Whether the question relies on the provided context, domain knowledge, or both.

Context-specific fields

The following fields appear where relevant to the configuration:

Field Description
image List of images accompanying the question. A single image is stored as a one-item list.
image_context Text associated with the images.
table Table content represented as a Markdown string.
table_context Text associated with the table.
image_N, image_context_N Individual image and its associated passage in the contextualized configuration.

Answer types

The answer_type field uses the following categories:

Value Answer format
numerical_value Numerical answer
numerical_value_with_units Numerical answer with units
equation_or_expression Equation or mathematical expression
short_answer Short textual answer
brief_explanation Brief explanation
long_explanation Extended explanation
calculation_with_work Calculation with working steps
multi_part_answer Answer containing multiple parts

The context_policy values are context_only, domain_knowledge_only, and context_first_then_domain_knowledge.

Available files

The dataset is provided in two formats:

  • Parquet: data/<configuration>/, with typed columns and embedded images. These files are used by the Hugging Face dataset viewer and load_dataset.
  • JSON Lines and images: subsets/<configuration>/<split>/, containing a metadata.jsonl file and the images referenced by its records.

Usage

Load a configuration using the Hugging Face datasets library:

from datasets import load_dataset

dataset = load_dataset(
    "SciKnowOrg/HighSchoolBench-OEQ",
    "context_image_only__text_response",
)

print(dataset)

Access a question and its reference answer:

example = dataset["test"][0]

print("Context:", example["context"])
print("Question:", example["question"])
print("Answer type:", example["answer_type"])
print("Reference answer:", example["answer"])

# Question images are decoded as PIL images.
images = example["image"]

List the available configurations:

from datasets import get_dataset_config_names

print(get_dataset_config_names("SciKnowOrg/HighSchoolBench-OEQ"))

To download the JSON Lines files and accompanying images:

hf download SciKnowOrg/HighSchoolBench-OEQ \
    --repo-type dataset \
    --include "subsets/*" \
    --local-dir HighSchoolBench-OEQ

Citation

If you use HighSchoolBench, please cite:

@misc{rani2026highschoolbench,
  title        = {HighSchoolBench: A Multimodal Benchmark of High School Assessment Questions},
  author       = {Rani, Krishna and D'Souza, Jennifer},
  year         = {2026},
  publisher    = {Hugging Face},
  howpublished = {\url{https://huggingface.co/datasets/SciKnowOrg/HighSchoolBench-MCQ},
                  \url{https://huggingface.co/datasets/SciKnowOrg/HighSchoolBench-OEQ}}
}

HighSchoolBench was introduced in the following master’s thesis:

@mastersthesis{rani2026highschoolbench_thesis,
  title  = {Toward Robust LLM Benchmarking in High School Science Education:
            Constructing a QA Corpus from Standardized Tests},
  author = {Rani, Krishna},
  school = {Gottfried Wilhelm Leibniz Universit{\"a}t Hannover},
  year   = {2026},
  doi    = {10.15488/22429},
  url    = {https://doi.org/10.15488/22429}
}
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