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What is the coefficient of $x^2y^6$ in the expansion of $\left(\frac{3}{5}x-\frac{y}{2}\right)^8$? Express your answer as a common fraction | To solve this problem, we will follow the steps outlined below:
## Step 1: Analyze the Problem Requirements
The problem asks us to find the coefficient of $x^2y^6$ in the expansion of the given binomial expression $\left(\frac{3}{5}x-\frac{y}{2}\right)^8$. This means we need to identify the term in the expansion that... | |
If a train travels 60 miles in 1.5 hours, what is its average speed in miles per minute? | Let's break down the problem step by step.
**Step 1: Analyze the problem requirements**
To solve this problem, we need to understand what information is given and what we're trying to find. The problem states that a train travels 60 miles in 1.5 hours, and we need to find its average speed in miles per minute.
Given... | |
How many 'a's are in the word banana? | To solve the problem of counting the number of 'a's in the word "banana," let's break down the solution into detailed steps, analyzing each part of the process thoroughly.
### Step 1: Analyze the Problem Requirements
The problem asks us to count the number of occurrences of the letter 'a' in the word "banana." To app... | |
A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost? | To solve this problem, let's break it down into steps as required.
### Step 1: Analyze the Problem Requirements
In this step, we need to understand what the problem is asking for. We are given two pieces of information:
1. The total cost of a bat and a ball together is $1.10.
2. The bat costs $1.00 more than the ball... | |
If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets? | Let's break down the problem step by step.
**Step 1: Analyze the problem requirements**
The problem states that 5 machines can make 5 widgets in 5 minutes. We need to determine how long it would take 100 machines to make 100 widgets. To approach this problem, we need to understand the relationship between the number ... | |
In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake? | Let's break down the problem step by step.
**Step 1: Analyze the problem requirements**
The problem states that there is a patch of lily pads in a lake that doubles in size every day. It takes 48 days for the patch to cover the entire lake. We need to find out how long it would take for the patch to cover half of the... | |
What is the next number in the sequence: 2, 6, 12, 20, 30, ...? | To solve this problem, let's follow the steps outlined below:
### Step 1: Analyze the Problem Requirements
The given sequence is: 2, 6, 12, 20, 30, ...
Our objective is to identify the pattern in this sequence and determine the next number.
Analysis:
- The sequence starts with 2 and increases by a certain amount ea... | |
If you have a 3-gallon jug and a 5-gallon jug, how can you measure exactly 4 gallons of water? | To solve this problem, let's follow the requirements step by step.
### Step 1: Analyze the Problem Requirements
The problem requires us to measure exactly 4 gallons of water using a 3-gallon jug and a 5-gallon jug. We don't have any other measuring tools or jugs available.
- **Given Information:**
- A 3-gallon ju... | |
A man is looking at a photograph of someone. His friend asks who it is. The man replies, 'Brothers and sisters, I have none. But that man's father is my father's son.' Who is in the photograph? | To solve this problem, let's follow the step-by-step approach outlined.
### Step 1: Analyze the Problem Requirements
The problem presents a scenario where a man is looking at a photograph, and when asked who the person in the photograph is, he replies with a cryptic message about family relationships. The key stateme... | |
Solve for x: 2x + 5 = 17 | To solve the equation 2x + 5 = 17, let's go through the steps as outlined:
### Step 1: Analyze the Problem Requirements
The problem is to solve for x in the linear equation 2x + 5 = 17. This means we need to find the value of x that makes the equation true. The given information is the equation itself, and the unknow... | |
What is the derivative of x^3 + 2x^2 - 5x + 1 with respect to x? | To find the derivative of the given function, x^3 + 2x^2 - 5x + 1, with respect to x, we will follow the step-by-step process outlined below.
### Step 1: Analyze the Problem Requirements
The problem asks for the derivative of a polynomial function with respect to x. The function given is x^3 + 2x^2 - 5x + 1. The deri... | |
A train is 100 meters long and travels at 100 meters per minute. How long will it take to completely pass through a 100-meter tunnel? | To solve this problem, we'll break it down into the steps as requested.
### Step 1: Analyze the problem requirements
The problem involves a train passing through a tunnel. We're given:
- The length of the train: 100 meters
- The speed of the train: 100 meters per minute
- The length of the tunnel: 100 meters
The obj... |
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