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, \(b = 40 − l = 40 − 20 = 20 m\).