Question

A rectangular sheet of fixed perimeter with sides having their lengths in the ratio $8:15$ is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is $100$ sq. units, the resulting box has maximum volume. Then the length of the side of the rectangular sheet is

• A. $40$
• B. $32$
• C. $45$
• D. $60$

Answer 1

The correct option is C $45$

Let sides of rectangular sheet be $8x$ and $15x$ where $x$ is fixed because perimeter of rectangular sheet is constant.
Let $t$ be the side of the square to be removed from all corners.


Now, $V\left(t\right)=(8x-2t)(15x-2t)t$
$=4t^3-46x{t}^2+120x^{2}t$
$V^{\prime }\left(t\right)=12t^2-92xt+120x^2{V}^{\prime }\left(t\right)=0\Rightarrow 3t^2-23xt+30x^2=0\Rightarrow (3t-5x)(t-6x)=0$
$t=6x$ $($reject as $t<4x)$
$\Rightarrow t=\frac{5}{3}x$
$V^{\prime \prime }\left(t\right)=24t-92x$
$V^{\prime \prime }\left(\frac{5}{3}x\right)<0$
Hence, volume is maximum at $t=\frac{5}{3}x$

Given that $4t^2=100$
$\Rightarrow t=5$
$\therefore x=3$
$\therefore$ Sides are $8x=8\times 3=24$ and $15x=15\times 3=45$