Question

A solid consists of a cone on top of a cylinder with a radius equal to that of the cone. The height of the cone is `66` and the height of the cylinder is `5`. If the volume of the solid is `64 pi`, what is the area of the base of the cylinder?

Answer 1

`A=64/27pi` `u^2`

Explanation:

The volume of the cone is given by: `v=1/3pir^2h`
Since the height of the cone is 66, then `h=66`
So, `v=1/3pir^2times66=22pir^2`

The volume of a cylinder is given by: `v=pir^2h`
Since the height of the cylinder is 5, then `h=5`
So, `v=pir^2h = pir^2times5=5pir^2`

The total volume of the solid is `64pi`
Therefore, `22pir^2+5pir^2=64pi`

`27pir^2=64pi`
`r^2=(64pi)/(27pi)`
`r^2=64/27`
`r=+-8/(3sqrt3)`
Since r is the radius, it must be have the restriction: `r>0`
Therefore, `r=8/(3sqrt3)`units

To find the base of the cylinder, we need to know that the base is a circle. The area of a circle is given by `A=pir^2 = pitimes(8/(3sqrt3))^2=64/27pi` `u^2`

Answer 2

The area of the base of the cylinder is: `A=pir^2=(64pi)/27`

Explanation:

The area of the base we need to find is: `A=pir^2`, where `r` is the radius of the cylinder.

The volume of the cylinder is: `pir^2*h_1`
where `h_1` is the height of the cylinder.
The volume of the cone is `pir^2*h_2/3`
where `h_2` is the height of the cone.

The volume of the solid is the sum of those two volumes: `V=pir^2*h_1 + pir^2*h_2/3`
Factoring `pir^2`:
`V= pir^2(h_1+h_2/3)`
`64pi = pir^2(5+66/3) = pir^2(5+22) = pir^2(27)`

`64pi = 27pir^2`

`pir^2 = (64pi)/27`
And that is the area of the base: `A=pir^2=(64pi)/27`