Question
A rectangular sheet of metal has dimensions . Equal squares of side cm are cut from each corner, and the sides are folded up to form an open box. Find for which the volume is maximum.
Solution — Step by Step
After folding, length = , width = , height = .
.
Divide by 12: . Solving: , giving or .
must be less than (since width ). So is rejected. Take .
. At : . So is indeed a maximum.
gives the maximum volume.
Why This Works
For an optimisation problem, we express the quantity to be maximised as a function of one variable, set the first derivative to zero, and check the second derivative for the type of extremum. The constraints (positive dimensions here) eliminate spurious critical points.
The cubic structure here is typical of “open-box” problems, which appear regularly in CBSE board exams.
Alternative Method
AM-GM inequality on — but the three factors are not symmetric, so we cannot apply AM-GM directly. Substitution and calculus is the cleanest path.
Common Mistake
Forgetting the constraint that all dimensions must be positive. comes out as a critical point but yields a negative width, so it is geometrically invalid. Always check that critical points satisfy the physical constraints of the problem.