Question

Is it possible to design a rectangular park of perimeter 80 m. and area 400 m2? If so, find its length and breadth.

Answer 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= 400[Given]
=> L² − 40L+ 400 = 0
Comparing this equation with aL² + bL + c = 0, we obtain
a = 1, b = −40, c = 400
Discriminant D = b² − 4ac = (− 40)² −4 (1) (400) = 1600 − 1600 = 0
As b² − 4ac = 0,
Therefore, this equation has equal real roots and hence, this situation is possible.
Root of this equation,

now , L² - 40L + 400= 0
L² - 20L - 20L + 400 = 0
L(L - 20) - 20(L - 20) = 0
(L - 20)(L - 20) = 0
L = 20m and B = 40 - L = 20m
therefore, length = 20m and Breadth =20m

Answer 2

Solution :

Given : Perimeter of a rectangular Park = 80 m

Area of a rectangular Park = 400 m²

Let the breadth of the rectangular Park = x m

Perimeter of a rectangular Park = 80 m

2(Length + breadth) = 80 m

Length + breadth = 80/2= 40 m

Length + breadth = 40 m

Length + x = 40  

Length of the rectangular Park = (40 - x) m

Area of rectangular Park=  length × breadth

A.T.Q

400 = (40 - x ) × x

x² - 40x + 400 = 0

x² - 20x - 20x +400 = 0

[ By factorization]

x (x -20) -20(x -20)= 0

(x - 20) (x - 20) = 0

(x - 20)= 0 or  (x - 20) = 0

x = 20 or x= 20

Breadth of a rectangular Park = x = 20 m

Length of a rectangular Park =(40 - x) = 40 -20 = 20 m.

Hence, it is possible to design the rectangular Park having perimeter 80 m and Area 400 m² of equal length and breadth i e 20 m each  

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