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 `42` and the height of the cylinder is `10`. If the volume of the solid is `225 pi`, what is the area of the base of the cylinder?

Answer 1

`29.452cm^2`

Explanation:

Lets assume we work in centimeters.
Top part of solid= a cone,bottom part a cilinder.
`=1/3pir^2h` = volume cone
`pir^2h`= vol cilinder
`1/3pir^2h+pir^2h=225pi`
multiply both sides with`1/pi`
`r^2(1/3h+h)=225`
`r^2(1/3*42+10)=225`
`24r^2=225`
`r^2=9.375`
substitute `r^2=9.375`
`1/3pir^2h+pir^2h=225pi`
`1/3*pi*9.375*42+pi*9.375*10=225pi`
`412.334+294.524=706.858`
`706.858=707.858`
Area base of cilinder=`pir^2`
`=3.141592654*9.375`
Area of base`=29.452cm^2`

Answer 2

`A = (75pi)/8`

`A = 29.45`

Explanation:

The volume of the cylinder and the cone together is `225 pi`

Let `h` = height of cylinder and `H` = height of cone.

Using the formulae gives:

`pi r^2 h + 1/3pi r^2 H = 225pi`

`pir^2 (10) + 1/cancel3pi r^2(cancel42^14) = 225 pi`

Note that we are not asked for radius, just for the area of the base of the cylinder which is given by `A = pir^2`

Solve for `pir^2`

`10 pir^2 + 14 pir^2 = 225pi`

`24pir^2 = 225pi`

`pir^2 = (225pi)/24`

`A = (75pi)/8`