Optimization Question

Question

Find the radius  and height  of a cylindrical can of total surface area  whose volume is as large as possible.

Answer 1

Volume V = πr²h

Surface area is missing here, so let K be a constant for the surface area.

A = 2πr² + 2πrh = K --> h = (K - 2πr²)/(2πr) --> h = (K-2πr²)/(2πr)

V = πr²(K-2πr²)/(2πr) = r(K-2πr²)/2 = rK/2 - πr³

Take the derivative of V and then set that to zero to solve for r

V ' = K/2 - 3πr² --> r = √(K/(6π)

h = (K-2π(K/6π))/(2πr) = (2/3)K/(2πr) = (K/3π)(1/√(K/(6π))

Max Volume = πr²h

Answer 2

Suppose surface area is A

Then A = 2πr² + 2πrh

h = (A-2πr²) / 2πr

V = πr²h = r (A-2πr²) / 2

dV/dr = (A - 6πr²) / 2

dV/dr = 0 gives r = √(A/6π)

You can find h by substituting the expression for r back into the equation for h