JEE Advanced · 2013 · 02 Jun 2013 · Paper 1 · Application of Derivatives

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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✓ Correct
(B)32
(C)45✓ Correct
(D)60

Answer: (A), (C)

Solution

Let the sides of the rectangular sheet be

8 a

and

15 a

. Squares of side

x

are removed from each corner, so

4 x^{2} = 100

, giving

x = 5

. The volume of the box is

V = (8 a - 2 x) (15 a - 2 x) x = (8 a - 10) (15 a - 10) (5)

. For maximum volume,

\frac{d V}{d x} = 0

. Differentiating and solving (treating

a

as the variable since

x = 5

is fixed), we get

a = 3

. Thus the sides are

8 (3) = 24

and

15 (3) = 45

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