A square piece of tin of side 30cm is to be made into a box without...

Given that, the side of square is $30cm$ 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,

$⇒ \dfrac{d V\left(x\right)}{d x} = 0$

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

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

On simplifying we get,

$⇒ x = 15cm, 5cm$

But $x$ cannot be $15cm$ as the volume becomes zero.

Hence $x = 5cm .$

Now to find the surface area of the cuboid,

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

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

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

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