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 $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.)

Answer 1

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$