https://file.aiquickdraw.com/mathbot/file/1726612990591g21ywt3e.jpg

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!

The volume V of a cone is given by the formula V = 1/3πr^2h. Part A: Solve for the height h. Part B: Find the height of a cone with V = 32π cm^3 and radius r of 4 cm.

Learn how to solve for the height of a cone using the volume formula V = 1/3πr^2h. This problem walks through solving for h and finding the height when V = 32π cm^3 and r = 4 cm. Perfect for students in Grades 9-10 studying geometry and algebra.

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