"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."

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