The volume of a right cone is defined as V =pir^2h/3 What is the simplified form of

Question

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)

Answer 1

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:

\[ \boxed{\frac{\pi^3 r^6 h^3}{27}} \]

Answer 2

Wrong

Answer 3

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.

Answer 4

Unclear answer

Answer 5

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.