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by sanju09 » Wed May 02, 2012 3:14 am
A rectangle has a perimeter of 20 inches. Which of the following could be the area of this rectangle in square inches?

I. 8
II. 23
III. 26

(A) I, only
(B) II, only
(C) I and II, only
(D) II and III only
(E) I, II, and III



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by mathbyvemuri » Thu May 03, 2012 12:14 am
let length and breadth be x and y respectively.
perimeter = 20 => 2(x+y) = 20 => x+y = 10
Area = x*y = x(10-x)
Differentiation method:
Area = f(x) = x(10-x) = 10x-x^2
first find first derivative of f(x):
f'(x) = 10-2x
second derivative, f''(x) = -2 < 0, hence f(x) has a maximum value at 'x' when f'(x) = 0
=> 10-2x = 0 => x = 5;
The maximum value of f(x) is at x=5 and is f(5) = 5(10-5) = 25 units
The maximum possible area is 25
Hence option III, which is 26 is ruled out. And options I and II are possible.
Answer "C"
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