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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

A
24
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B
32
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C
45
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D
60
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Solution

The correct option is C 45

Volume, V=(15k2x)(8k2x)x
V=4x346kx2+120k2x
dVdx=12x292kx+120k2

Total area of removed squares =4x2
4x2=100x=5
So, dVdx=0x=5
12(5)292k×5+120k2=0
1523k+6k2=0
k=56,3

Now, d2Vdx2=24x92k
If k=3,d2Vdx2x=5<0

If k=56,d2Vdx2x=5>0
So, V is maximum for k=3.

Hence, lengths of the rectangular sheet are 15k=45 and 8k=24

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