Asked by Mucho
The volume of a right cone is defined as V =pir^2h/3 What is the simplified form of
the expression that represents the cube of this volume, V^3? Treat the constant pi as you would a variable in your answer. (1 point)
the expression that represents the cube of this volume, V^3? Treat the constant pi as you would a variable in your answer. (1 point)
Answers
Answered by
GPT-4o mini
Given the volume of a right cone represented by the formula:
\[ V = \frac{\pi r^2 h}{3} \]
To find the cube of this volume, \( V^3 \), we need to cube the entire expression for \( V \):
\[
V^3 = \left(\frac{\pi r^2 h}{3}\right)^3
\]
Now, we will apply the cube to each part of the expression:
\[
V^3 = \frac{(\pi r^2 h)^3}{3^3}
\]
Calculating \( 3^3 \):
\[
3^3 = 27
\]
Now, for the numerator:
\[
(\pi r^2 h)^3 = \pi^3 (r^2)^3 h^3 = \pi^3 r^6 h^3
\]
Putting it all together, we have:
\[
V^3 = \frac{\pi^3 r^6 h^3}{27}
\]
Thus, the simplified form of the expression that represents the cube of the volume, \( V^3 \), is:
\[
\boxed{\frac{\pi^3 r^6 h^3}{27}}
\]
\[ V = \frac{\pi r^2 h}{3} \]
To find the cube of this volume, \( V^3 \), we need to cube the entire expression for \( V \):
\[
V^3 = \left(\frac{\pi r^2 h}{3}\right)^3
\]
Now, we will apply the cube to each part of the expression:
\[
V^3 = \frac{(\pi r^2 h)^3}{3^3}
\]
Calculating \( 3^3 \):
\[
3^3 = 27
\]
Now, for the numerator:
\[
(\pi r^2 h)^3 = \pi^3 (r^2)^3 h^3 = \pi^3 r^6 h^3
\]
Putting it all together, we have:
\[
V^3 = \frac{\pi^3 r^6 h^3}{27}
\]
Thus, the simplified form of the expression that represents the cube of the volume, \( V^3 \), is:
\[
\boxed{\frac{\pi^3 r^6 h^3}{27}}
\]
Answered by
GPT-4o mini
Let's go through the calculation step by step again to ensure we find the correct expression for \( V^3 \).
Starting with the volume of a right cone:
\[
V = \frac{\pi r^2 h}{3}
\]
To find \( V^3 \), we need to cube this volume:
\[
V^3 = \left(\frac{\pi r^2 h}{3}\right)^3
\]
When we cube this expression, we can distribute the exponent as follows:
\[
V^3 = \frac{(\pi r^2 h)^3}{3^3}
\]
Calculating \( 3^3 \):
\[
3^3 = 27
\]
Now, we cube the numerator \( (\pi r^2 h)^3 \):
\[
(\pi r^2 h)^3 = \pi^3 (r^2)^3 h^3 = \pi^3 r^6 h^3
\]
So putting it all together gives:
\[
V^3 = \frac{\pi^3 r^6 h^3}{27}
\]
Therefore, the correct simplified form of \( V^3 \) is:
\[
\boxed{\frac{\pi^3 r^6 h^3}{27}}
\]
If this doesn't match what you expected, please provide further details about what you believe is incorrect or what answer you are looking for.
Starting with the volume of a right cone:
\[
V = \frac{\pi r^2 h}{3}
\]
To find \( V^3 \), we need to cube this volume:
\[
V^3 = \left(\frac{\pi r^2 h}{3}\right)^3
\]
When we cube this expression, we can distribute the exponent as follows:
\[
V^3 = \frac{(\pi r^2 h)^3}{3^3}
\]
Calculating \( 3^3 \):
\[
3^3 = 27
\]
Now, we cube the numerator \( (\pi r^2 h)^3 \):
\[
(\pi r^2 h)^3 = \pi^3 (r^2)^3 h^3 = \pi^3 r^6 h^3
\]
So putting it all together gives:
\[
V^3 = \frac{\pi^3 r^6 h^3}{27}
\]
Therefore, the correct simplified form of \( V^3 \) is:
\[
\boxed{\frac{\pi^3 r^6 h^3}{27}}
\]
If this doesn't match what you expected, please provide further details about what you believe is incorrect or what answer you are looking for.
Answered by
GPT-4o mini
I apologize for any confusion caused. Let’s clarify the solution step-by-step to ensure everything is clear regarding cubing the volume of a right cone.
### Step 1: Volume of a Right Cone
The formula for the volume \( V \) of a right cone is given by:
\[
V = \frac{\pi r^2 h}{3}
\]
### Step 2: Cube the Volume
To find the cube of the volume, \( V^3 \), we take the expression for \( V \) and cube it:
\[
V^3 = \left( \frac{\pi r^2 h}{3} \right)^3
\]
### Step 3: Apply the Power
When cubing the entire fraction, we can separate it into parts:
\[
V^3 = \frac{(\pi r^2 h)^3}{3^3}
\]
Calculating \( 3^3 \):
\[
3^3 = 27
\]
### Step 4: Cube the Numerator
Now, we cube the numerator \( (\pi r^2 h)^3 \):
\[
(\pi r^2 h)^3 = \pi^3 (r^2)^3 h^3
\]
Breaking this down further:
- \( (r^2)^3 = r^{2 \cdot 3} = r^6 \)
Putting it all together:
\[
(\pi r^2 h)^3 = \pi^3 r^6 h^3
\]
### Step 5: Combine the Results
Now, substituting back into the expression for \( V^3 \):
\[
V^3 = \frac{\pi^3 r^6 h^3}{27}
\]
### Final Answer
Thus, the simplified expression for the cube of the volume \( V^3 \) is:
\[
\boxed{\frac{\pi^3 r^6 h^3}{27}}
\]
If there is something specific about this process that still seems unclear, please let me know, and I will do my best to address that directly.
### Step 1: Volume of a Right Cone
The formula for the volume \( V \) of a right cone is given by:
\[
V = \frac{\pi r^2 h}{3}
\]
### Step 2: Cube the Volume
To find the cube of the volume, \( V^3 \), we take the expression for \( V \) and cube it:
\[
V^3 = \left( \frac{\pi r^2 h}{3} \right)^3
\]
### Step 3: Apply the Power
When cubing the entire fraction, we can separate it into parts:
\[
V^3 = \frac{(\pi r^2 h)^3}{3^3}
\]
Calculating \( 3^3 \):
\[
3^3 = 27
\]
### Step 4: Cube the Numerator
Now, we cube the numerator \( (\pi r^2 h)^3 \):
\[
(\pi r^2 h)^3 = \pi^3 (r^2)^3 h^3
\]
Breaking this down further:
- \( (r^2)^3 = r^{2 \cdot 3} = r^6 \)
Putting it all together:
\[
(\pi r^2 h)^3 = \pi^3 r^6 h^3
\]
### Step 5: Combine the Results
Now, substituting back into the expression for \( V^3 \):
\[
V^3 = \frac{\pi^3 r^6 h^3}{27}
\]
### Final Answer
Thus, the simplified expression for the cube of the volume \( V^3 \) is:
\[
\boxed{\frac{\pi^3 r^6 h^3}{27}}
\]
If there is something specific about this process that still seems unclear, please let me know, and I will do my best to address that directly.
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