m.auld64 3 years ago 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

1. m.auld64

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

2. kumar2006

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.

3. m.auld64

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

4. Chlorophyll

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