Question

I need help! It is urgent. How is the volume of a hemisphere related to its circumscribed cylinder? Explain

Answer 1

A cylinder that circumscribers a hemisphere will share a base with the hemisphere. If the base has radius
`r`, then the volume of the hemisphere is

`V_(\rm hemisphere)=\frac12\left(\frac43\pi r^3\right)=\frac23\pi r^3`

A cylinder with radius
`r` and height
`h` has volume
`\pi r^3h`, but a cylinder that circumscribes a hemisphere will have a height equal to the radius of the hemisphere, so
`h=r`. Such a cylinder would have a volume of

`V_(\rm cylinder)=\pi r^3`

So, the volume of the hemisphere is
`\frac23` of the volume of the cylinder.