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 `33` and the height of the cylinder is `13`. If the volume of the solid is `256 pi`, what is the area of the base of the cylinder?

A Conical Volume is given by:
`V = 1/3 * pi * r^2 * h`
A Cylindrical Volume is given by:
`V = pi * r^2 * h`
Circular Area (base of cylinder)
`A = 2*pi * r^2`
Total solid volume =
`256pi = 1/3 * pi * r^2 * h_1 + pi * r^2 * h_2` Solve for r.
`256 = 1/3 * r^2 * 33 + r^2 * 13` ; `256 = r^2 * 11 + r^2 * 13`
`256 = r^2 * 24` ; `256/24 = r^2` ; `r^2 = 10.67 ;`

`A = pi * 10.76~~33.52`

Let's consider the diagram

We need to find the area of the base of the cylinder, which is a circle. The area of a circle is given by

`color(blue)("Area of circle"=pir^2`

Where `r` is the radius of the circle. We need to find `r^2`

`~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~`

The total volume of the solid is `256pi`

Therefore,

`color(purple)("Volume of cone"+"Volume of cylinder"=256pi`

We use the formulas

`color(orange)("Volume of cone"=1/3pir^2`

`color(orange)("Volume of cylinder"=pir^2h`

Where, `h` is the height and `r` is the radius. Let's put everything in the equation,

`rarr1/3pir^2h+pir^2h=256pi`

`rarr1/3pir^2*33+pir^2* 13=256pi`

`rarr1/(cancel3^1)pir^2*cancel33^11+pir^2* 13=256pi`

`rarrpir^2*11+pir^2*13=256pi`

`rarr24pir^2=256pi`

`rarr24cancelpir^2=256cancelpi`

`rarr24r^2=256`

`rarrr^2=256/24`

`color(violet)(rArrr^2=10.56`

`~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~`

Now, Let's find the area of the base

`color(green)("Area"=pir^2=pi*10.76~~33.52`