This is positive when x is positive, so the minimum area occurs when x = . Thus, the overall
dimensions of the poster are 4 + inches by 8 + 2 inches.
PROBLEM 3. An open-top box with a square bottom and rectangular sides is to have a volume of 256
cubic inches. Find the dimensions that require the minimum amount of material.
Answer: First, make a sketch of the situation.
The amount of material necessary to make the box is equal to the surface area.
S = x^2 + 4xyThe formula for the volume of the box is x^2 y = 256.
If we solve the latter equation for y, y = , and plug it into the former equation, we get
Now take the derivative and set it equal to zero.
If we solve this for x, we get x^3 = 512 and x = 8. Solving for y, we get y = 4.
Check that these dimensions give us a minimum.