Calculus word problem?

Question

The manager of Habagat Industries wishes to make a closed box with the following specifications. The box must have square base and a volume of 64 cubic centimeters. Find the dimensions for box with the smallest total surface of the package material costs P10.00 per square cm. Find the cost of the box.

Answer 1

Let base length of square base is x 

Volume of box  = x²h = 64 cm³

h = 64/ x²

Surface area of a box with square base is (A) = 4xh + 2x²

A = 4x(64/ x²) +2x²

A = 256/x +2x²

To find minimum value of dimensions for box we have to equate the first derivative to zero ( A'=0 )

A' = -256/x² +4x = 0

⇒ -256 + 4x³ = 0

⇒ 4x³ = 256 

⇒ x³=64

⇒x = 4

The surface area is = 256/4 + 2*16 ⇒ 96 cm²

So total cost of the box is = 96*10 = 960