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:

• Correct answer
  A

  $24$
• B

  $32$
• Correct answer
  C

  $45$
• D

  $60$

Solution

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Let $l = 8x,b = 15x$
$\therefore$ Volume$=\left(8x-2a\right)\left(15x-2a\right)\left(a\right)=4a^3-46a^{2}x+120a{x}^2$
$\frac{dV}{da} = 6a^2-46ax+60x^2$
${\left(\frac{dV}{da}\right)}_{at a = 5} = 0$
$\therefore x = 3$ and $\frac{5}{6}$
$\frac{d^{2}V}{d{a}^2} = 6a-23x$
${\left(\frac{d^{2}V}{d{a}^2}\right)}_{at a = 5\&x = 3}<0$
So, at $x=3$, it gives maxima:
${\left(\frac{d^{2}V}{d{a}^2}\right)}_{at a = 5\&x = \frac{5}{6}}>0$
So, at $\text{x} = \frac{5}{6}$ , it gives minima.
So by $x=3$ (for max volume):
$8x = 24, 15x = 45$

Hence, the correct answer is options: 'A', 'C'.