Question

part a
choose the formula for the volume of a cone $v = \frac{1}{3}\pi r^{2}h$ written in terms of $h$.
a. $h = \frac{r^{2}}{3v\pi}$
b. $h = \frac{v\pi r^{2}}{3}$
c. $h = \frac{\pi r^{2}}{3v}$
d. $h = \frac{3v}{\pi r^{2}}$
part b
find the height $h$ of a cone with volume $v = 32\pi\text{ cm}^3$ and radius $r = 4\text{ cm}$.
height = cm

Explanation:

Step1: Isolate h in volume formula

Given $V = \frac{1}{3}\pi r^{2}h$, multiply both sides by 3 to get $3V=\pi r^{2}h$. Then divide both sides by $\pi r^{2}$ to obtain $h=\frac{3V}{\pi r^{2}}$.

Step2: Solve for h in Part B

Substitute $V = 32\pi$ and $r = 4$ into $h=\frac{3V}{\pi r^{2}}$. First, calculate the denominator $\pi r^{2}=\pi\times4^{2}=16\pi$. Then $3V = 3\times32\pi=96\pi$. So $h=\frac{96\pi}{16\pi}=6$.

Answer:

Part A: D. $h=\frac{3V}{\pi r^{2}}$
Part B: 6