Expert: Ahmed Salami - 1/7/2010

Question
a cylindrical can without a top is made to contain V cm^3
of liquid. Find the dimensions that will minimize the cost
of the metal to make the can.  

Answer
Hi Patty,
If the cylinder with radius r and height h should contain a specific volume V
V = πr²h
h = V/πr²
The amount of material needed to make the can depends on the total surface area, which for an open cylinder is
A = 2πrh + πr²
 = 2πr(V/πr²) + πr²
 = 2V/r + πr²
Now we need to find the value of r that minimizes A, this occurs when dA/dr = 0
dA/dr = -2V/r² + 2πr
equating to zero,
-2V/r² + 2πr = 0
2πr = 2V/r²
r³ = V/π
r = ³√(V/π)
and
from 2πr = 2V/r²
πr = V/r²
r = V/πr²
 = h   (as can be seen at the top)
Therefore, the requires dimensions are
r = h = ³√(V/π)

You can always get back to me.

Regards