Question

A rectangular sheet of the 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 the equal area from all four corners. If the total area of removed squares is $100$, the resulting box has maximum volume. The lengths of the sides of the rectangular sheet are

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

Answer 1

Answer: (A), (C)

$45$

Explanation for the correct option:

Step 1. Find the length of the side of rectangular sheet:

Let the side of the square be $x$.

Volume $=(15k–2x)(8k–2x)x$

$=f\left(x\right)$

Now, $4\times x^2=100$

$\Rightarrow$ $x^2=25$

$\Rightarrow$ $x=5$

${\left(\frac{dV}{dx}\right)}_{x=0}=6x^2–46kx+60k^2$

Step 2. $f\text{'}\left(x\right)={\left(\frac{dV}{dx}\right)}_{x=0}$

$\Rightarrow$ $6x^2–46kx+60k^2=0$

$\Rightarrow$$6\times {\left(5\right)}^2–46k\left(5\right)+60k^2=0$

$\Rightarrow$ $15–23k+6k^2=0$

$\Rightarrow$ $6k^2–23k+15=0$

$\Rightarrow$ $k=3,\frac{5}{6}$

$\therefore$Side of the rectangle $=15k,8k$

$=15\times 3,8\times 3$

$=\mathbf{}\mathbf{45}\mathbf{,}\mathbf{}\mathbf{24}$

Hence, Option ‘A’ and ‘C’ is Correct.