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Question
Is it possible to design a rectangular park of perimeter 80 and area 400 m2? If so find its length and breadth.
Solution 1
Let the length and breadth of the park be l and b.
Perimeter = 2 (l + b) = 80
l + b = 40
Or, b = 40 - l
Area = l × b = l(40 - l) = 40l - l2 40l - l2 = 400
l2 - 40l + 400 = 0
Comparing this equation with al2 + bl + c = 0, we get
a = 1, b = -40, c = 400
Discriminant = b2 - 4ac
(-40)2 - 4 × 400
= 1600 - 1600 = 0
b2 - 4ac = 0
Therefore, this equation has equal real roots. And hence, this situation is possible.
Root of this equation, l = `-b/(2a)`
l = `(40)/(2(1))`
= `40/2`
l = 20
Therefore, length of park, l = 20 m
And breadth of park, b = 40 - l = 40 - 20 = 20 m.
Solution 2
Let the breadth of the rectangle be = x meters. Then
Perimeter = 80 meters
2(length + breadth) = 80
(length + x) = 40
length = 40 - x
And area of the rectangle
length × breadth = 400
(40 - x)x = 400
40x - x2 = 400
x2 - 40x + 400 = 0
x2 - 20x - 20x + 400 = 0
x(x - 20) - 20(x - 20) = 0
(x - 20)(x - 20) = 0
Yes, it is possible.
Hence, breadth of the rectangular park be 20 meters and length be 20 meters.
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