applying polynomial functions to volumes

**xian791** · September 25th 2008, 05:06 PM

I'm having trouble solving this problem. I don't know which radius they are looking for as well. The one in the buttom or the one on the side.

A silo is to be built in the shape of a right circular cylinder with a hemispherical top. If the total height of the silo is 30 feet and the total volume is 1008(pie) cubic feet, find the radius.

**Jhevon** · September 25th 2008, 06:03 PM

Originally Posted by xian791

I'm having trouble solving this problem. I don't know which radius they are looking for as well. The one in the buttom or the one on the side.

A silo is to be built in the shape of a right circular cylinder with a hemispherical top. If the total height of the silo is 30 feet and the total volume is 1008(pie) cubic feet, find the radius.

the total volume is given by the volume of the hemisphere and the volume of the cylinder. thus, if we let $V$ be the total volume

$V = \underbrace{\pi r^2 h}_{\text{volume of cylinder}} + \underbrace{\frac 23 \pi r^3}_{\text{volume of hemisphere}}$

where $h$ is the height of the cylinder and $r$ is the radius (both the cylinder and hemisphere have the same radius)

$\Rightarrow 1008 \pi = \pi r^2 h + \frac 23 \pi r^3$

now the key is, the total height is 30 feet. note that this height includes the radius of the hemisphere (which is the same as the radius of the cylinder). so that the height of the cylinder is $30 - r$ , thus we have

$1008 \pi = \pi r^2 (30 - r) + \frac 23 \pi r^3$

now solve for $r$

**xian791** · September 25th 2008, 06:48 PM

thanks man. i got it now

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