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The problem provided asks to find the height of a cone given its volume \( V = 32\pi \, \text{cm}^3 \) and radius \( r = 4 \, \text{cm} \).

The formula for the volume of a cone is:

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

where:
- \( V \) is the volume,
- \( r \) is the radius of the base, and
- \( h \) is the height of the cone.

### Step 1: Plug in the known values

We know that \( V = 32\pi \, \text{cm}^3 \) and \( r = 4 \, \text{cm} \), so the equation becomes:

\[
32\pi = \frac{1}{3} \pi (4)^2 h
\]

### Step 2: Simplify the equation

First, simplify \( (4)^2 \):

\[
32\pi = \frac{1}{3} \pi (16) h
\]

Next, cancel \( \pi \) from both sides:

\[
32 = \frac{1}{3} \times 16 h
\]

Now, simplify further:

\[
32 = \frac{16}{3} h
\]

### Step 3: Solve for \( h \)

Multiply both sides by 3 to eliminate the fraction:

\[
96 = 16 h
\]

Now, divide both sides by 16:

\[
h = \frac{96}{16} = 6 \, \text{cm}
\]

### Final Answer: 
The height of the cone is \( 6 \, \text{cm} \).

---

Would you like a more detailed explanation, or do you have any further questions?

Here are 5 related questions for practice:
1. What is the volume of a cone if the height is 9 cm and the radius is 5 cm?
2. How does changing the height of a cone affect its volume, assuming the radius is constant?
3. If the volume of a cone is \( 48\pi \, \text{cm}^3 \) and the radius is 3 cm, what is the height?
4. Derive the formula for the height of a cone in terms of volume and radius.
5. What is the surface area of a cone with a radius of 4 cm and a height of 6 cm?

**Tip:** Always check if the units are consistent in geometry problems to avoid calculation mistakes.

Find the height of a cone with V = 32π cm³ and radius r = 4 cm.

Learn how to find the height of a cone given its volume of 32π cm³ and a radius of 4 cm. This geometry problem involves calculating the height using the formula for the volume of a cone, V = (1/3)πr²h, and simplifying the equation. Suitable for students in Grades 8-10.

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