Proving that a cylinder's volume is equal to the sum of volumes of a sphere and cone

Question

I have to prove that a cylinder with the same height and width as a sphere and cone has the same volume as both the sphere and cone put together. Figured out that I had to put the formula into some algebra but not sure how to simplify it. $$ \pi r^2 h = \frac{4}{3} \pi r^3 + \frac{1}{3} \pi r^2 h $$

Answer 1

You need to show $$ \pi r^2 h = \frac{4}{3} \pi r^3 + \frac{1}{3} \pi r^2 h $$ Cancel $\pi r^2$ from both sides (assuming $r \ne 0$) to get $$ h = \frac{4r}{3} + \frac{h}{3} $$ which implies $h = 2r$.

So if $r=0$ (in which case all volumes are $0$) or if $h = 2r$, you have the relationship you seek.