Question

Is it possible to design a rectangular park of perimeter 80 m and area 400 $m^2$ ? If so, find its length and breadth.

Answer 1

Let the length and breadth of the park be $l$ and $b$.

Perimeter = $2 (l + b) = 80l + b = 40$
Or, $b = 40 − l$

Area = l$$ × b = l (40 − l)= 40l − l^2
$40l − l^2 = 400$
$l^2 − 40l + 400 = 0$

Comparing this equation with $al^2 + bl + c = 0,$
we obtain a = 1, b = −40, c = 400

Discriminate =$b^2 − 4ac = (− 40)^2 −4 (1) (400) = 1600 − 1600 = 0$
As $b^2 − 4ac = 0$, Therefore, this equation has equal real roots. And hence, this situation is possible.

Root of this equation,
$l = -\frac{b}{2a}$

$l = -\frac{(-40)}{2(1)} = \frac{40}{2} = 20$

Therefore, length of park, $l = 20 m$
And breadth of park, $b = 40 − l = 40 − 20 = 20 m$.