The volume of a cone with a radius of 7 cm is [tex]147 \pi[/tex] cubic centimeters. Which expression can be used to find the height [tex]h[/tex] of the cone?

A. [tex]147 \pi = \frac{1}{3}(7)(h)^2[/tex]

B. [tex]147 \pi = \frac{1}{3} \pi\left(7^2\right)(h)[/tex]

C. [tex]147 \pi = \frac{1}{3} \pi h[/tex]

D. [tex]147 \pi = \frac{1}{3} \pi(7)(h)[/tex]

To solve this problem, you need to find the correct expression for the height [tex]$ h $[/tex] of a cone given its volume and radius.

The formula for the volume of a cone is:

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

Given:
- Volume [tex]$ V = 147\pi $[/tex] cubic centimeters
- Radius [tex]$ r = 7 $[/tex] cm

We need to find an expression for [tex]$ h $[/tex].

Start with the formula for the volume of a cone:

[tex]\[
147\pi = \frac{1}{3} \pi (7^2) h
\][/tex]

Let's simplify it step-by-step:

1. Square the radius [tex]$ r = 7 $[/tex]:

[tex]\[
7^2 = 49
\][/tex]

2. Substitute [tex]$ 49 $[/tex] back into the equation:

[tex]\[
147\pi = \frac{1}{3} \pi \times 49 \times h
\][/tex]

3. Notice that we can divide both sides of the equation by [tex]$\pi$[/tex] (since [tex]$\pi$[/tex] is a common factor):

[tex]\[
147 = \frac{1}{3} \times 49 \times h
\][/tex]

4. Multiply both sides by 3 to eliminate the fraction:

[tex]\[
441 = 49 \times h
\][/tex]

5. Finally, solve for [tex]$ h $[/tex] by dividing both sides by 49:

[tex]\[
h = \frac{441}{49}
\][/tex]

Thus, the correct expression to find the height [tex]$ h $[/tex] is:

[tex]\[
147\pi = \frac{1}{3} \pi (7^2) h
\][/tex]

The second option: [tex]$ 147 \pi = \frac{1}{3} \pi(7^2)h $[/tex] is the correct expression for finding the height of the cone.