A storage shed is to built in the shape of a box with a square base. It should have volume 150ft^3 . The concrete for the base costs $4.00 per square foot, the roof material costs $2.00 per square foot, and the material for the sides costs $2.50 per square foot. Find the dimensions of the most economical shed. (Hint: the total cost will be equal the sum of the costs of each piece.)

Question

A storage shed is to built in the shape of a box with a square base. It should have volume 150ft^3 . The concrete for the base costs $4.00 per square foot, the roof material costs $2.00 per square foot, and the material for the sides costs $2.50 per square foot. Find the dimensions of the most economical shed. (Hint: the total cost will be equal the sum of the costs of each piece.)
thanks a lot

mathhelper · Answer

Let the base be x by x, and let the height be y feet
x^2 y = V
x^2 y = 150
y = 150/x^2

cost = 4x^2 + 2x^2 + 4(2.5)(xy)
= 6x^2 + 100x(150/x^2)
= 6x^2 + 15000/x
d(cost)/dx = 12x - 15000/x^2 = 0 for a min of cost
12x = 15000/x^2
x^3 = 1250
x = cuberoot(1250) = appr 10.77
then y = 150/x^2 = 1.29

state your conclusion

check: vol = (10.77)^2 (1.29) = 149.69 , close enough to 150