A square piece of tin of side 30   cm is to be made into a box without top by cutting a square from…

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

The required diagram is,

Now the dimensions of the cuboid formed will be 

$l\left(x\right)=30-2x$, $b\left(x\right)=30-2x$ and $h\left(x\right)=x$.

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

Now to get Maximum value,

$\Rightarrow \frac{dV\left(x\right)}{dx}=0$

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

$\Rightarrow \left(30-2x\right)\left(-4x+30-2x\right)=0$

On simplifying we get, 

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

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

Hence $\mathrm{x}=5 \mathrm{cm}.$

Now to find the surface area of the cuboid,

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

$=\left(30-2\times 5\right)\times 5\times 4+{\left(30-2\times 5\right)}^2$

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

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