JEE Main 2023 — Application Of Derivatives Question with Solution

Given, length of square sheet of side is $30\mathrm{cm}$.



Let, side of small square be $x\mathrm{cm}$.

Since, a square piece of tin is to be made into a box without top by cutting square from each corner and folding up the flaps to from a box. Then, this shape formed a cuboidal shape.

Thus, volume of the box $(V)$

$$\begin{array}{rl}& =(30-2x)(30-2x)x\\ & \\ & \Rightarrow V=x(30-2x{)}^2\end{array}$$

On differentiating both side with respect to $x$, we get

$$\begin{array}{rl}\frac{dV}{dx}& =(30-2x{)}^2+x\{2(30-2x)\cdot (-2)\}\\ & \\ & =(30-2x{)}^2-4x(30-2x)\\ & \\ & =(30-2x)\{30-6x\}\end{array}$$

Therefore, maximum of $V,\frac{dV}{dx}=0$

$$\begin{array}{ll}& \Rightarrow (30-2x)(30-6x)=0\\ & \\ & \Rightarrow x=15\text{ or }x=5[x=15\text{ (not taken) }]\end{array}$$

Thus, $x=5$

So, the surface area

$$\begin{array}{ll}=4\times (30-2x)\times x+(30-2x{)}^2& \\ & \\ =4\times (30-10)\times 5+(30-10{)}^2& (\text{ Put }x=5)\\ & \\ =400+400=800& \end{array}$$