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A running track of 440 ft. is laid out enclosing a football field, the shape of which is rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum, find the length of its longest sides.

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110

220

140

answer is B.

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Detailed Solution

Suppose,
Rectangle length = x.
Its breadth = y
Then perimeter of rectangle,

P = 2 x + π (\frac{y}{2}) + π (\frac{y}{2})

440 = 2 x + π \cdot y

y = \frac{440 - 2 x}{π}

Area of rectangular field portion,

A = x \cdot y

A = \frac{440 x - 2 x^{2}}{π}

Differentiate it with respect to x,

440 - 4 x = 0

x = 110 ft

Correct option is 2.

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