High School

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 [tex]h[/tex], the height 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} sh[/tex]

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

Answer :

The volume of a cone is given by the formula

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

Here, we are given that the volume is [tex]$147 \pi$[/tex] cubic centimeters and the radius is [tex]$7$[/tex] cm. Substituting these into the formula, we have

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

Since [tex]$7^2 = 49$[/tex], the equation becomes

[tex]$$
147\pi = \frac{1}{3} \pi (49) h.
$$[/tex]

This equation is exactly the second expression provided:

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

To verify, we can cancel [tex]$\pi$[/tex] from both sides:

[tex]$$
147 = \frac{49}{3} h.
$$[/tex]

Multiplying both sides by [tex]$\frac{3}{49}$[/tex] gives

[tex]$$
h = \frac{147 \times 3}{49} = 9.
$$[/tex]

Thus, the correct expression that can be used to find [tex]$h$[/tex] is:

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