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 $2/ft2, and the top is to be constructed of glass costing $4/ft2. what are the dimensions of the box of greatest volume that can be constructed for $72? (enter the dimensions (in feet) as a comma separated list.)

Box has all sides same leinght (a). Area of that box is 6*a^2.

Cost of material for bottom and 4 sides is 2$/ft^2. Area of bottom and sides is A=5*a^2

Cost of material for top is 4$/ft^2. Area of top of the box is T=a^2

A*2+T*4=72

5*a^2*2+a^2*4=72

a^2 * (10+4) = 72

a^2*14=72, a^2=72/14, a=root (72/14), a=2.267ft

Dimensions of the box: leinght of sides 2.2ft. Area of hole box 30.835ft^2

Area of the bottom and sides: A=5*a^2=25.696ft^2

Area of the top : T=a^2=5.139ft^2

Check: A*2$+T*4$=71.95$