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 `60` and the height of the cylinder is `15`. If the volume of the solid is `7 pi`, what is the area of the base of the cylinder?

Answer 1

The area of the base of the cylinder is `(7pi)/100`.

Explanation:

The volume of the solid is found by adding the volume of the cylinder to the volume of the cone. The volume of a cylinder is given by `V = pir^2h`. The volume of the cone is given by `V = (pir^2h)/3`. So the volume of the entire solid is given by:

`V = pir^2h + (pir^2h)/3`

This can be transformed by the converse of the Distributive Property to become:

`V = pir^2h(1 + 1/3)`

Which then becomes:

`V = (4pir^2h)/3`

Since we are looking for the area of the base, which is a circle, we need to isolate the part of the formula which gives that area. The area of a circle is given by `A = pir^2`, so we need to isolate that part of the simplified volume formula.

`3*V = 3*(4pir^2h)/3`
`3V = 4pir^2h`
`(3V)/(4h) = (4pir^2h)/(4h)`
`(3V)/(4h) = pir^2`

Therefore, the area of the base of the cylinder is given by the formula:

`A = (3V)/(4h)`

Substitute the values you know for `V` and `h` and simplify as possible. {`h = 60 + 15 = 75`}

`A = (3*7pi)/(4*75)`
`A = (21pi)/300`
`A = (7pi)/100`

Since the volume in the problem is left in terms of `pi`, it is acceptable to give the area of the base in terms of `pi` as well.