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 `18` and the height of the cylinder is `1`. If the volume of the solid is `49 pi`, what is the area of the base of the cylinder?

Answer 1

`color(blue)(7pi)`

Explanation:

We can use the volume of a cone, and the volume of a cylinder to solve this:

Volume of a cone is given as:

`V=1/3pir^2h_1`

Where `bb(h_1)` is cone height:

Volume of a cylinder is given as:

`V=pir^2h_2`

`bb(h_2)=` cylinder height.

We know the height of the cone and we know the height of the cylinder. We just need to find the radius, since this is common to both the cone and the cylinder.

Total volume of solid:

`V=pir^2h_2+1/3pir^2h_1=49pi`

Plugging in `h_1=18` and `h_2=1`

`pir^2(1)+1/3pir^2(18)=49pi`

`pir^2(1)+6pir^2=49pi`

`7pir^2=49pi`

`r^2=(49pi)/(7pi)=7`

Area of cylinder base:

`pir^2=7pi`