A rectangular sheet of fixed perimeter with sides having their length in the ratio 5 : 8 is converted into an open rectangular box by folding, after removing squares of equal areas from all four corners. If the total area of removed squares is 4, the resulting box has maximum volume. If the length of sides of rectangular sheet are a and b, then which of the following is/are TRUE? ([.] is GIF, a > b)

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[b] = 4

[a] = 8

[b] = 5

[a] = 9

answer is B, C.

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Detailed Solution

Let sides of rectangular be 8x, 5x and side of removed square be y.

$V = (8 x - 2 y) (5 x - 2 y) y V = 40 x^{2} y - 26 x y^{2} + 4 y^{3} \frac{d V}{d y} = 40 x^{2} - 52 x y + 12 y^{2} 4 y^{2} = 4 \Rightarrow y = 1 \frac{d V}{d y} = 0 \Rightarrow x = y, \frac{3}{10} y \frac{d^{2} V}{d y^{2}} = - 52 x + 24 y \frac{d^{2} V}{d y^{2}} < 0 @ x = y S i d e s a r e 8 y, 5 y .$