"a box is constructed out of two different types of metal. the metal for the top and bottom, which are both square, costs $4 per square foot and the metal for the sides costs $7 per square foot. find the dimensions that minimize cost if the box has a volume of 30 cubic feet."
Accepted Solution
A:
The base of the box is b ft by b ft and the height is h ft. Since the volume is 40 ft³, h = 40/b²
area of bottom = area of top = b² ft² area of one side = bh cost C = 2·4b² + 4·7bh = 8b² + 28b(40/b²) = 8b² + 1120/b dC/db = 16b - 1120/b²
min cost when dC/db = 0 16b = 1120/b² b = ∛70 ≅ 4.12 ft h = 40/b² ≅ 2.36 ft