A container with a square base, vertical sides, and open top is to be made from 200ft^2 of material. Find the dimensions (length of base and the height) of the container with greatest volume. If appropriate, leave your answer in radical form and enter all fractions in lowest terms.

Question

A container with a square base, vertical sides, and open top is to be made from 200ft^2 of material.  Find the dimensions (length of base and the height) of the container with greatest volume.  If appropriate, leave your answer in radical form and enter all fractions in lowest terms.

oobleck · Answer

If the base has sides x, and the height is h, then the area is
x^2 + 4xh = 200
so, h = (200-x^2)/(4x) = 50/x - x/4
Now the volume is
v = x^2 h = x^2 (50/x - x/4) = 50x - x^3/4
Now just find where dv/dx = 0 for maximum volume.
Post your work if you get stuck