Maximizing dimensions of a can with minimum cost but no given relation of cost

Question

A cylinder has a given volume of 1 cubic meter. The cost of constructing top and bottom of the cyl. is twice the cost of constructing the sides. What are the dimensions of the most economical can?

I have to find the maxima for dimensions depending on cost, but since there is no relation between the cost and the area, I am lost here. It isn't working if I take cost to be x/meter square.

Ans. given : radius $= \sqrt[3]{\frac1{4\pi}}$ m, height $= \sqrt[3]{\frac{16}{\pi}}$ m

Answer 1

HINT

From the givens, we have the following set up

• top and bottom area $A_1=2\pi R^2$
• lateral area $A_2=2\pi RH$
• total cost to minimize $C=2kA_1+kA_2=4k\pi R^2+2k\pi RH=2k\pi(2R^2+RH)$

with the constraint

• $V=\pi R^2H=1$