A closed box with square base is to be built to house an ant colony. The bottom of the box and all four sides are to be made of material costing 1 dollar/sq ft, and the top is to be constructed of glass costing 5 dollar/sq ft. What are the dimensions of the box of greatest volume that can be constructed for 72 dollars?

NOTE: Let denote the length of the side of the base and denote the height of the box

Question

A closed box with square base is to be built to house an ant colony. The bottom of the box and all four sides are to be made of material costing 1 dollar/sq ft, and the top is to be constructed of glass costing 5  dollar/sq ft. What are the dimensions of the box of greatest volume that can be constructed for 72 dollars?

NOTE: Let  denote the length of the side of the base and  denote the height of the box

anonymous · Answer

i know how to do this but im having trouble coming to the correct answer

anonymous · Answer

V = x^2 y ....... x = base

we also know that
72 = 5(x^2) + 4(xy) + x^2

72 = 6x^2 + 4xy
xy = (36 - 3x^2)/2

thus
V = x(36 - 3x^2)/2


get the derivative and set it to zero, the resulting x-value is the length of the base of the box with the greatest volume.

anonymous · Answer

can you help me do that cuz im apparently not doing it right

anonymous · Answer

Surface Area = 2 tops + 4 sides 
Area of square = 2 * x² + 4 * xy 
Total Cost:  $5 *x² + $1* ( 4xy + x²) = $ 72
                      6 x² + 4xy                   =    72
=>Area:                     xy   =   (72 - 6x² ) / 4
                                         =  18 - 1.5 x²
=> Volume:           x ( xy) =  18x - 1.5 x³
Thus maximum Volume: V' = 18 - 4.5 x² = 0   
                                           => x = 2 ft,  y = 6 ft