JEE Advanced | 2013Application of DerivativeHard
Question
A rectangular sheet of fixed perimeter with sides having their lengths in the ratio of 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
Solution
Where P = 8 λ + 15 λ + 8 λ +15 λ & λ is constant
Let removed length from each sides is x
Removed area is 4x2 = 100 ⇒ x = 5
V = (8λ - 2x) (15 λ - 2x)x
V = 120l2x - 46λ x2 + 4x3
12λ2 - 92λx + 12x2 = 0
Put x = 5 ⇒ 120λ2 - 4601 + 300 = 0
12λ2 - 40 λ + 30 = 0
6λ2 - 23λ + 15 = 0
( l - 3) (6λ - 5) = 0
= - 29λ + 24x = 120 - 92λ
at = 3
at
(rejected)
Let removed length from each sides is x
Removed area is 4x2 = 100 ⇒ x = 5
V = (8λ - 2x) (15 λ - 2x)x
V = 120l2x - 46λ x2 + 4x3
12λ2 - 92λx + 12x2 = 0Put x = 5 ⇒ 120λ2 - 4601 + 300 = 0
12λ2 - 40 λ + 30 = 0
6λ2 - 23λ + 15 = 0
( l - 3) (6λ - 5) = 0
= - 29λ + 24x = 120 - 92λ at = 3
at
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