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

Question

A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in cm$^2$) is equal to :

Accepted Answer

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{aligned}

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

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

$$
\begin{aligned}
\frac{d V}{d x} & =(30-2 x)^2+x\{2(30-2 x) \cdot(-2)\} \\
& =(30-2 x)^2-4 x(30-2 x) \\
& =(30-2 x)\{30-6 x\}
\end{aligned}
$$

Therefore, maximum of $V, \frac{d V}{d x}=0$

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

Thus, $x=5$

So, the surface area

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

Solutions · 1