Question
A rectangular sheet of cardboard 24 cm by 18 cm has equal squares cut from its four corners. The sides are then folded up to form an open-top box. Find the side length of the cut squares that maximizes the volume of the box. (CBSE-pattern problem — appeared in 2022.)
Solution — Step by Step
After cutting squares of side from each corner and folding, the box has:
- length
- width
- height
Volume:
Expanding:
Set to zero:
So cm or cm.
The valid range is (since width ). So cm.
. At : . Maximum confirmed.
cm gives maximum volume.
Why This Works
Optimization with constraint: the volume is a cubic in , so gives two critical points. The physical constraint rules out the larger root. The second derivative test confirms which is a maximum.
This problem template — “open box from a rectangular sheet” — appears in CBSE board exams and JEE Main almost every other year. Memorize the three-step pattern: write as a function, set , check second derivative.
For optimization, always state the domain of the variable. A cubic has two critical points, but only one usually lies in the feasible region. The other is mathematical noise.
Alternative Method
For a square sheet of side , the optimal cut is . For a non-square sheet, no clean closed form exists — solve the quadratic numerically. Some shortcut formulas in coaching books assume a square sheet — don’t trust those for rectangular cases.
Common Mistake
Students take both roots of as valid answers, or pick the wrong one. Always check the domain: the cut depth must be less than half the shorter side ( cm here), so is geometrically impossible.