Question

A sphere and the base of a cone have a radius of 3 inches. The volume of the sphere equals the volume of the cone. what is the height of the cone, in inches?

Answer 1

So the volume of a sphere is $V= \frac{4}{3} \pi r^3$ , and the volume of a cone is $V= \pi r^2 \frac{h}{3}$ (h=height, r=radius). Knowing only the radius, we can solve for the volume of the sphere. (Also I'm going to be leaving answers in pi form.)

$V= \frac{4}{3} \pi 3^3$
Solve the exponents to get $V= \frac{4}{3} \pi 9$
Multiply 4/3 and 9 to get $V=13.5 \pi$


Now that we know the volume of the sphere, we can solve for the height of the cone since the volumes for both are the same.

$13.5 \pi = \pi 3^2 \frac{h}{3}$

Solve the exponents to get $13.5 \pi =9 \pi \frac{h}{3}$

Divide $9 \pi$ on each side to get $1.5= \frac{h}{3}$

Multiply 3 on each side, and your answer should be $4.5=h$