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Question:

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

Solution:

Let the sides of the rectangle be 15k and 8k, and the side of the square be x. Then (15k - 2x)(8k - 2x)x is the volume.
→v = 2(2x³ - 3kx² + 60k²x)
For maximum volume, the derivative of V w.r.t. x should be zero.
→dv/dx|x=5 = 0
6x² - 6kx + 60k²|x=5 = 0
6k² - 6k + 15 = 0
k = 3, k = 5/6
Only k = 3 is permissible. So, the sides are 45 and 24.