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 $100$, the resulting box has maximum volume. Then the lengths of the sides of the rectangular sheet are

• A. $24$
• B. $32$
• C. $45$
• D. $60$

Answer 1

Answer: (A), (C)

$45$


Volume, $V=(15k-2x)(8k-2x)x$
$\Rightarrow V=4x^3-46k{x}^2+120k^{2}x$
$\frac{dV}{dx}=12x^2-92kx+120k^2$

Total area of removed squares $=4x^2$
$\therefore 4x^2=100\Rightarrow x=5$
So, $\frac{dV}{dx}=0\Rightarrow x=5$
$\therefore 12(5{)}^2-92k\times 5+120k^2=0$
$\Rightarrow 15-23k+6k^2=0$
$\Rightarrow k=\frac{5}{6},3$

Now, $\frac{d^{2}V}{d{x}^2}=24x-92k$
If $k=3,\frac{d^{2}V}{d{x}^2}{|}_{x=5}<0$

If $k=\frac{5}{6},\frac{d^{2}V}{d{x}^2}{|}_{x=5}>0$
So, $V$ is maximum for $k=3$.

Hence, lengths of the rectangular sheet are $15k=45$ and $8k=24$