A rectangular box with a square base and no top is to contain exactly 54 cubic centimeters. Find the dimensions that will yield the minimum cost if the base costs 4 times as much as the sides.
I am not sure how to apply Lagrange Multiplier Method to this problem as my teacher asked. I started with creating the function $C(s,h) = s^2 + 4hs$ for the cost of the box with base $s$ and height $h$. The restraint is $g(s,h) = s^2h = 54$. Is this the right setup? Thanks!
Not quite. If the base sides have length s and the height is h, then the base has area $s^2$ and each side has area sh. There are 4 sides so the sides have total area 4sh. The surface area of the box is $s^2+ 4sh$. However, you are not asked to minimize the surface area. You are asked to minimize the cost. Since the base cost four times as much as the sides, the cost will be $4s^2+ 4sh$. The volume is $s^2h$ and the "constraint" is, as you say, $s^2h= 54$.