Let's solve the problem step by step.

Part A:

The volume $V$ of a cone is given by the formula:

$V = \frac{1}{3} \pi r^2 h$

We need to solve for the height $h$. Rearranging the equation to isolate $h$:

$h = \frac{3V}{\pi r^2}$

So, the correct option for Part A is Option D:

$h = \frac{3V}{\pi r^2}$

Part B:

We are asked to find the height of a cone with:

• Volume $V = 32\pi \, \text{cm}^3$
• Radius $r = 4 \, \text{cm}$

We will use the formula from Part A:

$h = \frac{3V}{\pi r^2}$

Substitute the given values:

$h = \frac{3 \times 32\pi}{\pi \times 4^2}$

Simplifying:

$h = \frac{96\pi}{\pi \times 16} = \frac{96}{16} = 6 \, \text{cm}$

So, the height of the cone is 6 cm.

Do you want any further details or explanations?
Here are five related questions:

1. How is the volume of a cylinder related to the volume of a cone?
2. What happens to the height if the radius of a cone is doubled, keeping the volume constant?
3. How would you calculate the slant height of a cone if the height and radius are known?
4. How can you use integration to derive the formula for the volume of a cone?
5. How is the surface area of a cone calculated?

Tip: Always double-check units when working with volume and area to ensure consistency in your calculations!