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Alex used to practice yoga 2 times a week but decided to add an extra 3 days a week. She practices yoga for 45 minutes in the morning and 30 minutes in the evening every day she practices. How many minutes a week does she practice yoga now? | nvidia_math |
What is the truncation of the product of the partition generating function \( P(x) \) and the modified partition generating function \( P^*(x) \) to degree \( 10 \)? Specifically, calculate the polynomial consisting of all terms in the power series expansion of \( P(x)P^*(x) \) with degree less than or equal to \( 10 \... | automath |
Auction Mechanism Design Problem:
Suppose you are the owner of a vintage car, and you want to auction it off. You have four interested buyers, and you know each buyer's maximum price for the car. The potential buyers' maximum prices for the car are $60,000, $70,000, $80,000, and $85,000.
How can you design an auctio... | tiger_lab_math |
the length of a rectangle is one fourth of the radius of a circle . the radius of the circle is equal to the side of the square , whose area is 784 sq . units . what is the area ( in sq . units ) of the rectangle if the rectangle if the breadth is 5 units ? Please respond by writing a program in Python. | tiger_lab_math |
The base of a right pyramid is a regular pentagon with side length 4 units. The altitude of the pyramid is 6 units. If the pyramid is transformed into a cube by folding along its edges, find the side length of the resulting cube. | nvidia_math |
Suppose there is a company with 100 employees and the probability of any given employee committing financial fraud is 0.05. What is the probability that at least 5 employees in the company will commit financial fraud? | tiger_lab_math |
Define $f(x) = \frac{x + 6}{x}.$ The sequence $(f_n)$ is given by $f_1 = f$ and $f_n = f \circ f_{n - 1}$ for $n \ge 2.$ For example, $f_2(x) = f(f(x)) = \frac{7x + 6}{x + 6}$ and $f_3(x) = f(f_2(x)) = \frac{13x + 42}{7x + 6}.$ Let $S$ be the set of real numbers $x$ such that $f_n(x) = x$ for some positive integer $n.$... | tiger_math |
Find three-fifths of $\frac{12}{7}$. | nvidia_math |
Two trains of length 100 m and 200 m are 100 m apart. They start moving towards each other on parallel tracks, at speeds 54 kmph and 72 kmph. In how much time will the trains cross each other?
Answer Choices: (A) 80/9 sec (B) 80/7 sec (C) 30/7 sec (D) 80/8 sec (E) 81/7 sec | tiger_lab_math |
A small pool filled only with water will require an additional 300 gallons of water in order to be filled to 85% of its capacity. If pumping in these additional 300 gallons of water will increase the amount of water in the pool by 30%, what is the total capacity of the pool in gallons?
Answer Choices: (A) 1000 (B) 1250... | tiger_lab_math |
Grace started her own landscaping business. She charges $6 an hour for mowing lawns, $11 for pulling weeds and $9 for putting down mulch. In September she mowed lawns for 63 hours, pulled weeds for 9 hours and put down mulch for 10 hours. How much money did she earn in September? | tiger_math |
Under what conditions does a smooth fibration \( p: E \to B \) between compact smooth manifolds guarantee that the projection \( p \) is a submersion, and consequently, that the fibration is a fiber bundle? | automath |
if √ 2 n = 64 , then the value of n is : ? Let's program in Python in the response. | tiger_lab_math |
Rodney and Todd are rowing down a river that is 50 yards wide at one end. If the river's width increases from this end uniformly by 2 yards every 10 meters along, and they row along the river at a rate of 5 m/s, how long (in seconds) will it take them to get to the point where the river is 80 yards wide? | tiger_math |
Tom has decided to start a stamp collection. He buys a pack of stamps every week, with each pack containing one more stamp than the previous week's pack. The cost of each pack is proportional to the number of stamps it contains. If Tom spends a total of $120 on stamps over the course of 8 weeks, and the cost of the fir... | nvidia_math |
In the set $T = \{1, 2, 3, \ldots , 100\}$, how many integers are left after removing the multiples of 5 and the multiples of 7, but adding back the multiples of 35? | tiger_math |
"statement 1": If the average speed of the whole journey was 36 mph , then what is Tom ' s speed driving from B to C in miles per hour ?, "statement 2" :If the average speed of the whole journey was less than 36 mph , then what is Tom ' s speed driving from B to C in miles per hour ?, "options: " Entailment or contrad... | tiger_lab_math |
How many positive two-digit integers are multiples of 5 and of 7? Let's write a Python program. | tiger_lab_math |
Mark, a lawyer, goes to the county courthouse every day. It takes him 5 minutes to find parking and 3 minutes to walk into the courthouse. On 2 days of the week, it takes him 30 minutes to get through the metal detector, while on the other 3 days, it only takes him 10 minutes. If he works for 5 days in a week, how much... | tiger_math |
Given the parameters of a battery charge simulation, where the maximum capacity is \(10 \, kWh\), the minimum capacity is \(2 \, kWh\), and the initial battery level is \(5\, kWh\), how many iterations will it take for the battery to reach its maximum capacity \(10\, kWh\) if the charge levels per iteration are represe... | automath |
A square sheet of paper had squares of side 3 inches cut from each corner to form an open box with a volume of 675 cubic inches. What was the side length of the original square sheet of paper? | automath |
If the sum of two numbers is 45 and their difference is 3, what is the smaller of the two numbers? | tiger_math |
Benjamin went to McDonald's to buy something for dinner. He bought a salad, a burger, and two packs of fries. He paid in total $15. How much did the burger cost, if one pack of fries was for $2 and the salad was three times that price? | tiger_math |
Reena took a loan of $1200 with simple interest for as many years as the rate of interest. At the end of the loan period, she paid a certain amount as interest, and the rate of interest was 3%. How much interest did she pay? | tiger_math |
For which values of \( n \) does there exist at least one pair of nonconjugate even permutations in \( A_n \) that share the same cycle type, and what is the specific cycle type of such permutations for each of these values? | automath |
Ava bakes 48 cupcakes and 24 muffins. Ava eats 1 cupcake and 1 muffin. Ava's sister eats 2 cupcakes and 1 muffin. If Ava sells a cupcake for $1.25 and a muffin for $2, how much money will Ava make if she sells all of her baked goods? | nvidia_math |
Determine the coefficient of $x^2$ in the expansion of $(2x+3)^5$? | tiger_math |
If the speed of a man is 51 km per hour, then what is the distance traveled by him in 30 seconds ?
Answer Choices: (A) 275m (B) 360m (C) 375 m (D) 425 m (E) 440 m | tiger_lab_math |
Amon & Rhonda combined have 215 marbles. If Amon owns 55 marbles more than Rhonda, how many marbles does Rhonda have? | tiger_lab_math |
Given the matrix \( A = \begin{pmatrix} x & y \\ y & x \end{pmatrix} \) where \( x, y \in \mathbb{R} \), compute the matrix exponential \( \exp(A) \) and express your answer in the form \( \exp(x) \begin{pmatrix} a & b \\ b & a \end{pmatrix} \) for some functions \( a(y) \) and \( b(y) \). What are the specific forms o... | automath |
If a car has two front tires with a diameter of 10 inches, and two rear tires with a diameter of 12 inches, what is the least common multiple of the total number of revolutions made by each set of tires when the car travels 1/2 mile? | tiger_math |
What is half of the absolute value of the difference of the squares of 18 and 16? | nvidia_math |
Apple sold 100 iPhones at their New York store today for an average cost of x. They also sold 20 iPads for an average cost of $900 and 80 Apple TVs for an average cost of $200. What was the average cost across all products sold today?
If we know the answer to the above question is 670, what is the value of unknown vari... | tiger_math |
Mary went to the store to buy fruit. Apples cost $1, oranges cost $2, and bananas cost $3. For every 5 fruits that customers buy, the store offers a $1 discount. Mary buys 5 apples, 3 oranges, and 2 bananas. How much will she pay? | tiger_lab_math |
Rationalize the denominator of $\frac{3}{2\sqrt{7} - 3\sqrt{13}}$ and write your answer in the form $\displaystyle \frac{A\sqrt{B} + C\sqrt{D}}{E}$, where $B < D$, the fraction is in lowest terms and all radicals are in simplest radical form. What is $A+B+C+D+E$? | tiger_math |
In triangle ABC, angle A is 60 degrees, and angle B is 80 degrees. The circle centered at O is circumscribed about triangle ABC. What is the measure of angle BOC, in degrees? | nvidia_math |
Moses and Tiffany want to see who is faster. But instead of a race, they just start running down the block. Tiffany runs 6 blocks in 3 minutes. Moses runs 12 blocks in 8 minutes. What is the speed (defined as blocks per minute) of the runner with the higher average speed? | tiger_lab_math |
Given two linearly independent vector fields \( V_1 \) and \( V_2 \) on \( \mathbb{R}^3 \) such that their commutator \( [V_1, V_2] = 0 \) and they satisfy \( V_1 f = 0 \) and \( V_2 f = 0 \) for some function \( f: \mathbb{R}^3 \to \mathbb{R} \), what is the form of \( f \) in terms of its dependence on the spatial va... | automath |
What is the cumulative distribution function (CDF) and the 0.95 quantile of the standardized Student's-t distribution with 5 degrees of freedom? | automath |
The length of a rectangle-shaped book is 5 cm longer than the diameter of a round clock. How many cm is the diameter of the clock if the length of the book is 30 cm? | nvidia_math |
A plumber contractor decides to field-test two new pumps. One is rated at 180 gallons per hour and the other at 250 gallons per hour. Over a period of 6 hours , she pumps a total of 1325 gallons. How many hours were used by the 250 gallons per hour pump? Let's write a Python program. | tiger_lab_math |
Tina saved $27 in June, $14 in July, and $21 in August. Then Tina spent $5 on books and $17 on new shoes. How much money does Tina have left? | tiger_math |
In how many ways can I choose a 4-person committee from a club of 9 people? | tiger_lab_math |
Find all solutions to the equation $\!\sqrt{2-3z} = 9$. Express your answer as a common fraction. | nvidia_math |
What is the least positive three-digit multiple of 7? Let's write a program. | tiger_lab_math |
Let $a$, $b$, and $c$ be the real roots of
\[x^3 + px^2 + qx + r = 0.\]
Given that $a+b+c = -10$, $ab+bc+ca = 14$, and $abc = -24$, find the value of $p$. | nvidia_math |
What is the value of $$\frac{1}{2}\times4\times\frac{1}{8}\times16\times\frac{1}{32}\times64\times\frac{1}{128}\times256\times\frac{1}{512}\times1024?$$ Please respond by writing a program in Python. | tiger_lab_math |
Calculate the value of 50000 - ((37500 / 62.35)^2 + √324) = ? | tiger_math |
A function $f$ satisfies the condition $f(f(x)) = x^2$ for all real numbers $x$. If $f(x) = ax + b$, what is the value of $a$? | nvidia_math |
In our school's girls volleyball team, we have a total of 14 players, including a set of triplets named Alicia, Amanda, and Anna. How many different ways can we select 6 starters without any restrictions? (Note: The triplets are considered distinguishable.) | tiger_math |
A piece of paper in the shape of a rectangular parallelogram has two sides that are 6 cm long and two sides that are 8 cm long. Find the curvature of this surface using the Gauss-Bonnet theorem. | tiger_lab_math |
If Betty's account balance is $3,456 and each of Gina's two bank accounts contains a quarter of that balance, what is the total combined balance of both of Gina's accounts? | tiger_math |
Given a Monty Hall simulation with \( x \) doors, where a contestant randomly picks a door and the host reveals a goat behind one of the remaining doors, what is the probability of winning a car if the contestant always randomly chooses to switch to one of the other unopened doors? Express this probability in terms of ... | automath |
Given the definition of "posterity" with respect to the relation of immediate predecessor, what is the posterity of the number 5, and how does it differ from the concept of the successor? Please provide a clear mathematical explanation for your answer. | automath |
What is the $x$-intercept of the line perpendicular to the line defined by $3x-2y = 6$ and whose $y$-intercept is 2? | nvidia_math |
A group of friends want to paint a large mural on a wall. If 8 friends work together, they can finish the mural in 6 hours. However, if they work in two shifts of 4 friends each, with a 30-minute break in between shifts, how many total hours will it take to complete the mural? | nvidia_math |
Triangle $\triangle ABC$ has a right angle at $C$, $\angle A = 60^\circ$, and $BC=5$. Find the radius of the incircle of $\triangle ABC$. | tiger_math |
A clothing store has 10 different shirts, 8 different pants, and 5 different pairs of shoes. In how many ways can a customer choose one shirt, one pair of pants, and one pair of shoes? | tiger_lab_math |
A number was chosen and divided by 4. Then, 175 was subtracted from the result and the final answer was a certain value. The chosen number was 740. What was the final answer after subtracting 175? | tiger_math |
Abigail, Beatrice, and Carson combine their eggs to sell them at the market. If Abigail has 40 eggs, Beatrice has 52 eggs, and Carson has 17 eggs, and if eggs can only be sold in cartons of 12, how many eggs will be left over if all cartons are sold? | tiger_math |
Consider a linear homogeneous recurrence relation of the form \( a_n = c_1 a_{n-1} + c_2 a_{n-2} \) with characteristic polynomial \( P(x) = x^2 - c_1 x - c_2 \). If the roots of this polynomial are \( r_1 \) and \( r_2 \), express the general solution of the recurrence relation in terms of these roots. Additionally, i... | automath |
Determine whether the series \(\sum_{n=1}^{\infty}\frac {\sin{\frac 1 n}} {\sqrt n}\) converges or diverges. If it converges, provide a reasoning based on asymptotic behavior or comparison tests. | automath |
Tim sleeps x hours per day for 5 days a week and 10 hours a day for the other 2. How much total sleep does he get?
If we know the answer to the above question is 50, what is the value of unknown variable x? | tiger_math |
What is the value of the definite integral \(\int_0^{\infty} \frac{dx}{(a^2 + x^2)^7}\) expressed in terms of \(\pi\) and \(a\)? | automath |
How can we factor the polynomial \(1 + x^4\) into the product of two quadratic polynomials with real coefficients? Specifically, what steps lead to the factorization \((x^2 - \sqrt{2}x + 1)(x^2 + \sqrt{2}x + 1)\)? | automath |
Samara and x of her friends heard alligators spotted on the local river and decided to join a search organized by the wildlife service to capture the animals. After searching the whole day, Samara had seen 20 alligators while her friends had seen an average of 10 alligators each. Calculate the total number of alligator... | tiger_math |
Let $b_n = \frac{2^{2n+1}+1}{3}$. Define $e_n$ to be the greatest common divisor of $b_n$ and $b_{n+1}$. What is the maximum possible value that $e_n$ can take on? | nvidia_math |
What is the smallest prime number \( p \) such that the gap between \( p \) and the next prime number \( p' \) is greater than 10? | automath |
Daniel works at an electronics store, and he claims that the popularity of a toaster (measured in number of sales) is inversely proportional to its cost. If 12 customers buy a toaster that costs $\$500$, according to Daniel's theory, how many customers would buy a toaster that costs $\$750$? | nvidia_math |
Molly Roe believes her average daily balance for a 28-day billing cycle is $877.67. If the following transactions occurred during the cycle:
- Previous balance: $800
- 3/24: Payment of $60
- 3/29: Charge of $250 at Sears
- 4/5: Payment of $20
- 4/9: Charge of $200 at Macy's
What is the correct average daily bala... | automath |
A running track is the ring formed by two concentric circles. If the circumferences of the two circles differ by $10\pi $ feet, how wide is the track in feet?
[asy]size(100); path g=scale(2)*unitcircle;
filldraw(unitcircle^^g,evenodd+grey,black);
[/asy] | nvidia_math |
I have 5 groups of friends, each group having 3 people. I want to invite one person from each group to a party. However, there is one person who cannot be invited together with another specific person from a different group. How many possible combinations of guests can I invite to the party? | nvidia_math |
A cube has 8 vertices, and each vertex can be connected to every other vertex to form a triangle. However, some of these triangles are contained within the faces of the cube, while others pass through the interior of the cube. How many triangles can be formed using the vertices of the cube that pass through the interio... | nvidia_math |
The sum of the perimeters of two squares is 100. The area of a square is equal to the square of its side length. The sum of the areas of the two squares is 442, while the difference in their areas is 120. What is the length of the side of the larger square? | tiger_math |
The lengths of the three sides of a triangle are $11$, $y+6$ and $3y+2$. The perimeter of the triangle is 50. What is the length of the longest side of the triangle? | nvidia_math |
Ariana is past due on three bills. The first bill for $200 charges 10% simple interest for each overdue month and it's 2 months overdue. The second bill for $130 charges a flat $50 late fee per month and is 6 months overdue. The last bill is two months late and charges a $40 fee the first month overdue and twice that f... | tiger_math |
Let $a$ and $b$ be angles such that $\cos (a + b) = \frac{1}{3}$ and $\cos (a - b) = \frac{2}{3}.$ Find $\frac{\tan a}{\tan b}.$ | tiger_math |
Simplify $(r^2 + 3r - 2) - (r^2 + 7r - 5)$. | nvidia_math |
What is the Lyapunov exponent of the Rikitake system with parameter values a = 1, b = 2, c = 6? | tiger_lab_math |
Given that $r$, $s$, and $t$ are the roots of $2x^3 - 7x^2 - 6 = 0$, what is the product $rst$? | tiger_math |
The average expenditure of a labourer for 7 months was 85 and he fell into debt. In the next 4 months by reducing his monthly expenses to 60 he not only cleared off his debt but also saved 30. His monthly income is
Answer Choices: (A) 70.11 (B) 72.02 (C) 86.5 (D) 78.2 (E) None of the above | tiger_lab_math |
Given a control system with a feedback factor \( K_{FB} = 0.5 \) and an open loop gain \( T(s) \), what is the relationship between the crossover frequency \( f_c \) of the open loop gain and the 3dB corner frequency of the closed loop gain? If the crossover frequency \( f_c \) is known, what is the expression for the ... | automath |
This morning, farmer Rita's workers collected 30,000 gallons of milk and chilled it in a storage tank. They then spent the next 4 hours pumping milk at a rate of 2,880 gallons/hour from the storage tank into a milk tanker. For the next 7 hours, the workers added more milk into the storage tank, at a rate of 1,500 gallo... | nvidia_math |
Let a square pyramid have vertices $(0,0,0)$, $(3,0,0)$, $(0,3,0)$, $(3,3,0)$, and $(0,0,h)$.
Find the smallest possible volume of the pyramid if it contains the point $(1,1,1)$ and $h>0$. | nvidia_math |
Problem:
In the triangle $ABC$, the points $D$ and $E$ are the midpoints of the sides $AB$ and $AC$ respectively. The vectors $\mathbf{AD}$ and $\mathbf{AE}$ are orthogonal. If the length of the side $BC$ is $6$, find the length of the side $AB$. | nvidia_math |
Consider two 4-bit binary strings: \( A = 1101 \) and \( B = 1010 \). Calculate the Hamming distance between \( A \) and \( B \). | automath |
Two brick walls are being built. Each wall has 30 bricks in a single row and there are 50 rows in each wall. How many bricks will be used to make both walls? | tiger_math |
3/5 of all married couples have more than one child. 2/5 of all married couples have more than 3 children. What fraction of all married couples have 2 or 3 children?
Answer Choices: (A) 1/5 (B) 1/4 (C) 7/20 (D) 3/5 (E) It cannot be determined from the given information. | tiger_lab_math |
Given the integers \( x = 97929 \) and \( y = 83261 \), and the set of primes \( P = [1171, 1429, 1811, 2339] \), determine whether there exists a positive integer \( B \) of the form \( B = kx + y \) (for some integer \( k \)) such that \( B \) is divisible only by the primes in the set \( P \). If such a \( B \) exis... | automath |
The polynomial $2x^3 + bx + 7$ has a factor of the form $x^2 + px + 1.$ Find $b.$ | nvidia_math |
Determine the sum of all odd integers between 11 and 39, inclusive. | tiger_math |
In a potentiometer experiment, a cell exhibits a balancing length of 240 cm. When a resistance of 2Ω is shunted across the cell, the balancing length decreases to 120 cm. What is the internal resistance of the cell?
A) 1Ω
B) 0.5Ω
C) 2Ω
D) 4Ω | automath |
Shanna planted 6 tomato plants, 2 eggplant plants, and 4 pepper plants in her summer garden. Half of her tomato plants and one pepper plant died. The remaining plants each gave her 7 vegetables. How many vegetables did she have in all to harvest? Let's write a Python program. | tiger_lab_math |
A truck has its center of gravity located halfway between its front and rear wheels. When the front wheels are on a scale, it reads \(W_1\), and when the rear wheels are on the scale, it reads \(W_2\). If the weight of the truck can be calculated as \(W_1 + W_2\), what is the total weight of the truck if \(W_1 = 5000\)... | automath |
If $a$ is the smallest positive two-digit multiple of $3$, and $b$ is the smallest positive three-digit multiple of $4$, then find $a+b$. | tiger_math |
A cheerleader waves her pom-pom in simple harmonic motion with an amplitude of 18.0 cm and a frequency of 0.850 Hz. What is the time required for the pom-pom to move from the equilibrium position to a point 12.0 cm away from the equilibrium position? | automath |
Emily manages a juice stand at a local market. On a busy Saturday, she filled four large jugs with 3 liters of juice each, six medium bottles with 1 liter each, and eight small cups with 0.25 liters each. How many liters of juice did Emily sell in total? | nvidia_math |
Given two signals from a neuron response, \( S_1 \) acquired 60 times and \( S_2 \) acquired 4 times, where \( S_1 \) represents the response to stimulus A and \( S_2 \) the response to stimulus A+B, how should one process these signals statistically to isolate the response attributable to stimulus B? Specifically, if ... | automath |
Conor can chop 12 eggplants, 9 carrots, and 8 potatoes in a day. If he works x times a week, how many vegetables can he chop?
If we know the answer to the above question is 116, what is the value of unknown variable x? | tiger_math |
Roderick read that you should spend 15 minutes stretching for every 2 hours you spend watching TV. If Roderick spends 8 hours binge-watching his favorite show, how long will he spend stretching? | nvidia_math |
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