Question

A cylinder, cone, and sphere are shown below. The three figures have the same radius. The cylinder and cone have the same height with h = r. If the volume of the cone is 36 cubic units, what are the volumes of the cylinder and the sphere? Explain how you found your answers. Answer in complete sentences and include all relevant calculations.

Answer 1

Vcylinder=107.8

Vsphere=143.79

Explanation:

The first thing you should do is find the radius using the volume of the cone

Vcone=(1/3)pi h r^2

h=r

solving for r

`\sqrt[3]{((3Vcone)/(\pi ) )} =r\\`

r=[tex]\sqrt[3]{3*36/\pi } =r=3.25

Sphere=

V=(4/3)*pi*r^3

V=(4/3)*pi*(3.25)^3=143.79

cilinder

h=r

V=pi*r^3

V=pi*(3.25)^3=107.8

Answer 2

In this problem, we have three solids, namely:

1. A cylinder.

2. Cone.

3. sphere.

These three figures have the same radius. The cylinder and cone have the same height with
`h=r`. We also know that:

`Volume \ of \ the \ cone \ is \ 36units^3`

Also, some formulas are provided:

`Volume \ of \ a \ cylinder: V=\pi r^2h \\ \\ Volume \ of \ a \ cone: V=(1)/(3)\pi r^2h \\ \\ Volume \ of \ a \ sphere: V=(4)/(3)\pi r^3`

Since
`h=r` then our new formulas are:

`Volume \ of \ a \ cylinder: V=\pi r^3 \\ \\ Volume \ of \ a \ cone: V=(1)/(3)\pi r^3 \\ \\ Volume \ of \ a \ sphere: V=(4)/(3)\pi r^3`

For the cone, we know that
`V=36units^3` then we can get the radius
`r`:

`V=(1)/(3)\pi r^3 \\ \\ 36=(1)/(3)\pi r^3 \\ \\ 36(3)=\pi r^3 \\ \\ r^3=(108)/(\pi)`

VOLUME OF THE CYLINDER:

`V=\pi r^3 \\ \\ V=\pi \left((108)/(\pi)\right) \\ \\ \boxed{V=108units^3}`

VOLUME OF THE SPHERE:

`V=(4)/(3)\pi r^3 \\ \\ V=(4)/(3)\pi \left((108)/(\pi)\right) \\ \\ \boxed{V= 144units^3}`