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 `9` and the height of the cylinder is `15`. If the volume of the solid is `48 pi`, what is the area of the base of the cylinder?

Answer 1

See a solution process below:

Explanation:

The formula for the volume of a cylinder is:

`V_c = pir^2h_c`

The formula for the volume of a cone is:

`V_n = pir^2h_n/3`

We can now write an equation for the volume of the complete solid as:

`V_c + V_n = 48pi`

`pir^2h_c + pir^2h_n/3 = 48pi`

Where:

• `r` is the radius of the cone and the cylinder.
• `h_c` is the height of the cylinder
• `h_n` is the height of the cone

We can substitute what we know from the problem and solve for `r`:

`(pir^2 xx 15) + (pir^2 xx 9/3) = 48pi`

`15pir^2 + 3pir^2 = 48pi`

`(15 + 3)pir^2 = 48pi`

`18pir^2 = 48pi`

`(18pir^2)/(color(red)(18)color(red)(pi)) = (48pi)/(color(red)(18)color(red)(pi))`

`(color(red)(cancel(color(black)(18pi)))r^2)/cancel(color(red)(18)color(red)(pi)) = (48color(red)(cancel(color(black)(pi))))/(color(red)(18)cancel(color(red)(pi)))`

`(color(red)(cancel(color(black)(18pi)))r^2)/cancel(color(red)(18)color(red)(pi)) = (48color(red)(cancel(color(black)(pi))))/(color(red)(18)cancel(color(red)(pi)))`

`r^2 = 48/color(red)(18)`

`r^2 = (6 xx 8)/color(red)(6 xx 3)`

`r^2 = (color(red)(cancel(color(black)(6))) xx 8)/color(red)(color(black)(cancel(color(red)(6))) xx 3)`

`r^2 = 8/3`

`sqrt(r^2) = sqrt(8/3)`

`r = sqrt(8/3)`

The base of a cylinder is a circle. The formula for the area of a circle is:

`A = pir^2`

We can substitute the radius we calculate above to determine the area of the circle which is the same as the area of the base of the cylinder:

`A = pi(sqrt(8/3))^2`

`A = pi8/3`

`A = 8/3pi`