A rectangular sheet of fixed perimeter with sides having their...

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:

Options

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

Correct answer

A. $24$

Step-by-step solution

Let

l = 8x,b = 15x

∴

Volume

= \left(8x - 2a\right)\left(15x - 2a\right)\left(a\right) = 4{a}^{3} - 46{a}^{2}x + 120a{x}^{2}

\dfrac{dV}{da} = 6{a}^{2} - 46ax + 60{x}^{2}

{\left(\dfrac{dV}{da}\right)}_{at a = 5} = 0

∴ x = 3

and

\dfrac{5}{6}

\dfrac{{d}^{2}V}{d{a}^{2}} = 6a - 23x

{\left(\dfrac{{d}^{2}V}{d{a}^{2}}\right)}_{at a = 5&x = 3} < 0

So, at

x = 3

, it gives maxima:

{\left(\dfrac{{d}^{2}V}{d{a}^{2}}\right)}_{at a = 5&x = \dfrac{5}{6}} > 0

So, at

\text{x} = \dfrac{5}{6}

, it gives minima.
So by

x = 3

(for max volume):

8x = 24, 15x = 45

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