Question

5. 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

Answer

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 - l240l -  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 = 20

Therefore, length of park, l = 20 m

And breadth of park, b = 40 - l = 40 - 20 = 20 m.

Answer 2

let the length of the park be l and be its breadth.

given

l + b =40m

lb =200 m^2

NOW let us consider an equation with its roots as l and b

$\bold{ \large{it \: is \: of \: the \: form}} \\ \\ \bold{ \large{ {x}^{2} - (l + b)x + lb = 0}} \\ \\ \bold{ \large{ {x}^{2} - 40x + 200 = 0}} \\ \\ \bold{ \large{ {x}^{2} - 20x - 20x + 400 = 0}} \\ \\ \bold{ \large{x(x - 20) - 20(x - 20) = 0}} \\ \\ \bold{ \large{ {(x - 20)}^{2} = 0}} \\ \\ \bold{ \large{l = b = 20}}$

Therefore the rectangular park is a square with side length 20 m.

Hope it helps.