Expert: Alon Mandes - 7/29/2009

Question
A silo is to hold 10,000 cubic feet of grain. The silo will be cylindrical in shape and have a flat top. The floor of the silo will be the earth. What dimensions of the silo will use the least amount of material for construction.

I know that I have to set up the function, find the derivative and set it equal to 0 to find the minimum value of the dimensions. i do not know how to set up the equation. Any feedback will be greatly appreciated.

Answer
The silo shape is cylindrical . That means :
1. The surface area is S=2πrh+πr² ("1st term is the perimeter & 2nd term the roof")
2. The volume is V=πr²h
Our function to be minimized is S . But we need 1st to find connection between r & h.
To do so we use the given information : "A silo is to hold 10,000 cubic feet of grain".
That means : V=10000 --> πr²h=10000 --> h=10000/πr² . We substitute this result into eq 1 :
S=2πrh+πr²=2πr*10000/πr² + πr² .
S(r)=20000/r + πr².
S'(r)=-20000/r² + 2πr .
S'(r)=0 --> 2πr=20000/r² --> πr³=10000 --> r=14.71
Now let's find h :
h=10000/πr² = 10000/π(14.71)² = 14.71
The dimensions will be r=h=14.71

Alon.