An open‑top rectangular box is to be made from a sheet of material having total surface area $1200{\text{cm}}^2$. Find the dimensions (length $l$, width $w$, height $h$) that give the maximum possible volume. Give the dimensions correct to two decimal places.

Step-by-step solution

1. For an open box, surface area $S$ is $S=lw+2lh+2wh=1200$.
2. Volume $V=lwh$ is to be maximised.
3. Because the constraint is symmetric in $l$ and $w$, the maximum occurs when $l=w=a$ (square base).
4. Substitute $l=w=a$ into the area equation: $a^2+4ah=1200$, so $h=\frac{1200-a^2}{4a}$.
5. Express volume in terms of $a$: $V=a^{2}h=a^2\left(\frac{1200-a^2}{4a}\right)=\frac{a(1200-a^2)}{4}$.
6. Differentiate $V(a)$: $\frac{dV}{da}=\frac{1200-3a^2}{4}$.
7. Set derivative to zero: $1200-3a^2=0\Rightarrow a^2=400\Rightarrow a=20$cm (positive root).
8. Find $h$ using $h=\frac{1200-a^2}{4a}=\frac{1200-400}{80}=10$cm.
9. Thus the box of maximum volume has $l=w=20$cm and $h=10$cm.

Concept tested: Optimization – maximum volume with given surface area

Published 17 September 2026 · How this solution was prepared and checked