JEE Mathematics 2023 Question with Solution

Given: Let $x$ be the side length of the square cut from each corner of the square sheet.

Find: The surface area of the open box when its volume is maximum.

The dimensions of the open box are:

• Length = $30 - 2x$
• Breadth = $30 - 2x$
• Height = $x$

The volume is

To maximize the volume, differentiate:

Set $V'(x) = 0$:

Simplifying,

Factorizing,

So, $x = 5$ or $x = 15$.

If $x = 15$, then the length and breadth become $30 - 2(15) = 0$, which is not feasible. Therefore, $x = 5$.

Now the surface area of the open box without top is

Substituting $x = 5$,

the solution concludes the final value is $800 \, \text{cm}^2$, but it also marks the correct option as C. Since the solution working is the primary source and the listed options do not contain $800$ at option C, this is a source discrepancy. The defensible option by value is A.

From the critical points $x = 5$ and $x = 15$, only values with $0 < x < 15$ can form a box, because the base side $30 - 2x$ must remain positive. Thus $x = 15$ makes the base collapse to zero size, so it cannot give a valid box. Hence the maximum-volume box occurs at $x = 5$, and its surface area is $800 \, \text{cm}^2$.