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

$\text{sides ratio } 8:15, \quad \text{perimeter fixed}$

$\text{4 corner squares, total area } = 100$

$\text{box volume is maximum}$

$\text{To find: the side lengths of the sheet}$

$\text{sides } 8\lambda \text{ and } 15\lambda$

$\text{corner square side } x$

$V = x(8\lambda - 2x)(15\lambda - 2x)$

$V = 120\lambda^2 x - 46\lambda x^2 + 4x^3$

$\dfrac{dV}{dx} = 120\lambda^2 - 92\lambda x + 12x^2$

$12x^2 - 92\lambda x + 120\lambda^2 = 0 \Rightarrow 3x^2 - 23\lambda x + 30\lambda^2 = 0$

$x = 6\lambda \ \text{or} \ x = \dfrac{5\lambda}{3}$

$x = 6\lambda:\ 8\lambda - 2x = -4\lambda < 0 \ \Rightarrow \text{reject}$

$4x^2 = 100 \Rightarrow 4\left(\dfrac{5\lambda}{3}\right)^2 = 100 \Rightarrow \lambda = 3$

$8\lambda = 24, \quad 15\lambda = 45$

$\boxed{\text{Sides } 24 \text{ and } 45 \ \text{— options (1), (3)}}$