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
- 24
- 32
- 45
- 60
The Correct Option is C
Solution and Explanation
Let $l = 15\, x$ and $b = 8x$
Then Volume $= V = (8x - 2a)(15x - 2a).a$
$= 4a^3 - 46a^2x + 120ax^2$
$dV/da = 6a^2 - 46ax + 60x^2$
$d2V/da^2 = 12a - 46x$
Now, $dV/da = 0$ gives
$? 6a^2 - 46ax + 60x^2 = 0$
$? 30x^2 - 23ax + 3a^2 = 0$
$? 30x^2 - 18ax - 5ax + 3a^2 = 0$
$? 6x(5x - 3a) - a(5x - 3a) = 0$
$? (6x - a)(5x - 3a) = 0$
$? x = a/6, 3a/5$
$? x = 5/6 , 3$ when $a = 5$
When $x = 3$, $a = 5$, $d^2V/da^2 < 0$
So the Volume is maximum.
Hence, the lengths are $l = 15.3 = 45$ and $b = 8.3 = 24$.
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