Given the volume of a right cone $V = \frac{\pi r^2 h}{3}$, we need to find the simplified form of the expression that represents the cube of this volume, $V^3$.

Step 1: Write the volume expression.

The volume of the cone is: $V = \frac{\pi r^2 h}{3}$

Step 2: Cube the volume expression.

To find $V^3$, we raise the expression for $V$ to the third power: $V^3 = \left(\frac{\pi r^2 h}{3}\right)^3$

Step 3: Expand the cube.

When cubing a fraction, cube both the numerator and the denominator: $V^3 = \frac{(\pi r^2 h)^3}{3^3}$

Step 4: Simplify the numerator and the denominator.

Now, simplify each part:

• The denominator becomes $3^3 = 27$.
• The numerator becomes $(\pi r^2 h)^3 = \pi^3 (r^2)^3 h^3 = \pi^3 r^6 h^3$.

So, the simplified form of the expression is: $V^3 = \frac{\pi^3 r^6 h^3}{27}$

Final Answer:

The simplified form of the expression that represents the cube of the volume is: $V^3 = \frac{\pi^3 r^6 h^3}{27}$

Would you like any more details or have any questions?

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Here are 8 related questions for further exploration:

1. How does the volume formula change if the cone is a right circular cone with a different base shape?
2. What is the derivative of the volume of a cone with respect to the radius $r$?
3. How would you express the surface area of a cone in terms of $r$ and $h$?
4. How does the volume of a cone compare to the volume of a cylinder with the same base and height?
5. What is the physical interpretation of cubing the volume in real-world scenarios?
6. How would the expression for $V^3$ change if the cone had an elliptical base?
7. What is the effect on the volume if the height of the cone is doubled while the radius remains the same?
8. How would you compute the volume of a cone if the height is not perpendicular to the base?

Tip: When dealing with volume formulas, remember to check the units and ensure they are consistent throughout the calculation.