A manufacturer wants to design an open-top box having a square base and a surface area of 90 square inches. What dimensions will provide a box with maximum volume?

Question

A manufacturer wants to design an open-top box having a square base and a surface area of 90 square inches.  What dimensions will provide a box with maximum volume?

anonymous · Answer

Please help!!

anonymous · Answer

Let s and h be the square's edge length and the box height respectively.
Then:$$\left\{V\text{=}s^{2 }h,s^{2 }+4s h=90\right\}$$Solve the second equation for h and plug the result back in expression for V.
Can you take it from here?

anonymous · Answer

$$V\text{= }\frac{1}{4} s \left(90-s^2\right)$$The above equation expresses the box volume as a function of s, the square's side length.  A Plot is attached.

anonymous · Answer

thank you very much! @robtobey

anonymous · Answer

This problem can be solved with one of Mathematica's Controlled Optimization functions, Maximize.  No knowledge of the Calculus is necessary.

$$\text{Maximize}\left[\left\{s^{2 }h,s^{2 }+4s h=90,s>0,h>0\right\},\{s,h\}\right] $$$$\left\{15 \sqrt{30},\left\{s\to \sqrt{30},h\to \sqrt{\frac{15}{2}}\right\}\right\} $$