JEE Main 2023 — Mathematics Calculus

Given that, the side of square is $30\mathrm{cm}$30cm and $x\text{cm}$xcm squares are cut off.

The required diagram is,

Now the dimensions of the cuboid formed will be

$l(x)=30-2x$l(x)=30−2x, $b(x)=30-2x$b(x)=30−2x and $h(x)=x$h(x)=x.

The Volume of the cuboid will be $V(x)={(30-2x)}^{2}(x)$V(x)=(30−2x)2(x)

Now to get Maximum value,

$$\Rightarrow \frac{dV(x)}{dx}=0$$⇒dxdV(x)​=0

$$\Rightarrow 2(30-2x)(-2)x+{(30-2x)}^{2}(1)=0$$⇒2(30−2x)(−2)x+(30−2x)2(1)=0

$$\Rightarrow (30-2x)(-4x+30-2x)=0$$⇒(30−2x)(−4x+30−2x)=0

On simplifying we get,

$$\Rightarrow x=15\mathrm{cm},5\mathrm{cm}$$⇒x=15cm,5cm

But $x$x cannot be $15\mathrm{cm}$15cm as the volume becomes zero.

Hence $x=5\mathrm{cm}.$x=5cm.

Now to find the surface area of the cuboid,

Surface area will be $=(30-2x)\times x\times 4+{(30-2x)}^{2}$=(30−2x)×x×4+(30−2x)2

$$=(30-2\times 5)\times 5\times 4+{(30-2\times 5)}^{2}$$=(30−2×5)×5×4+(30−2×5)2

$=800{\mathrm{cm}}^{2}$=800cm2.

Therefore, the required surface area will be $800{\mathrm{cm}}^{2}$800cm2.