the surface area cost of construction per m^2 for a large box is $50 for the base, $60 for the top and $40 for the walls. Find the dimensions of this box that minimize the cost if its volume has to be 9m^3 and its height, 1m

Answer 1

let the base be x by y m
V = (x)(y)(1) = xy
xy = 9
y = 9/x

cost = 50xy + 60xy + 2(40x) + 2(40y)
= 110xy + 80x + 80y
= 110x(9/x) + 80x + 720/x
= 110 + 80x + 720/x

d(cost)/dx = 80 - 720/x^2
= 0 for a min of cost
80x^2 = 720
x^2 = 9
x = 3
then y = 9/3 = 3

the box should be 3 by 3 by 1 m