Expert: Ahmed Salami - 10/29/2009

Question
A carpenter has been asked to build an open box with a square base. The sides of the box will cost $3 per square meter, and the base will cost $4 per square meter. What are the dimensions of the box of greatest volume that can be constructed for $48?

Answer
Hi Tyler,
let the square base have sides with length x and the height be h.
The base area = x²
The box has four other sides with area xh and so the total area = 4xh
The total cost of the base is then 4.x² = 4x²
The total cost of the other four sides is 3.4xh = 12xh
If the total cost should equal 48, then
4x² + 12xh = 48
x² + 3xh = 12
3xh = 12 - x²
h = (12 - x²)/3x
Now, the volume of the box = base area . height
V = x².h
 = x².(12 - x²)/3x
 = x.(12 - x²)/3
 = (12x - x³)/3
 = 4x - x³/3
The greatest volume occurs when dV/dx = 0
dV/dx = 4 - x²
equating to zero,
4 - x² = 0
x² = 4
x = ±2
We disregard -2 and take x = 2
Then h = (12 - x²)/3x
      = (12 - 2²)/3(2)
      = (12 - 4)/6
      = 8/6
      = 4/3

Regards