A pencil cup with a capacity of 25π in.3 is to be constructed in the shape of a right circular cylinder with an open top. If the material for the side costs 5/8 of the cost of the material for the base, what dimensions should the cup have to minimize the construction cost?

Answer 1

πr^2 h = 25π
so, h = 25/r^2
The cost function
c = πr^2 + 5/8 * 2πrh = πr^2 + 5/4 πr*25/r^2 = π(r^2 + 125/(4r))
dc/dr = π(2r - 125/(4r^2))
dc/dr = 0 when 8r^3 = 125
so, minimum cost is at r = 5/2

Answer 2

What is the height value?