Question

A manufacturer wants to design an open box (no top) having a square base and
a surface area of 300 square inches. What dimensions will produce a box with
maximum volume?

Answer 1

Calculus problem?

Answer 2

Let b = length if one side of square base
Let h = height of box
Surface area = 300 = b^2 + 4hb
Height is found by rearranging as follows:
\[h=\frac{300-b ^{2}}{4b}\]
Let volume = V
\[V=b ^{2}h=b ^{2}\times \frac{300-b ^{2}}{4b}=75b-\frac{b ^{3}}{4}\]
Now we have the volume as a function of b. The maximum value of V is found by differentiation.

Answer 3

\[V=75b-\frac{b ^{3}}{4}\]
\[\frac{dV}{db}=?\]

Answer 4

Very nice work. To finish, take the derivative with respect to b to get\[V'=75-\frac{3b^2}{4}=0 \implies b=10\]