Express the amount of Area "A" of material to make the can as a function of the radius "r" of the cylinder, then determine for what values of "r" is the Area "A" the smallest

Question

A can in the shape of a right circular cylinder is required to have a volume of 250 cubic centimeters. Express the amount of Area "A" of material to make the can as a function of the radius "R" of the cylinder, then determine for what values of "r" is the Area "A" the smallest.

Answer 1

A=2πRh+2πR² is the total area of the can including the circular ends, where h is the height.

The volume is πR²h=250cc, so h=250/(πR²).

Therefore A in terms of r is given by:

A=(2πR)(250/(πR²)+2πR²=500/R+2πR².

We can differentiate this:

dA/dR=-500/R²+4πR=0 at an extremum.

So, -500+4πR³=0, and R³=500/(4π)=125/π.

And R=5/∛π=3.4139cm approx.

(Substitute this value for R and A=219.6887 sq cm, whereas if we take a value of R slightly more or less than 3.4139cm, we get a value for A bigger than 219.6887 sq cm, so R=3.4139cm gives minimum surface area.)

 

Note that h=2R for this minimum condition and that a cross-section of the can through the diameters of the top and bottom form a perfect square, so this question could have been solved without calculus, because volume=250=2πR³.