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 \(b\), respectively.

The perimeter of the park is given by \(2 (l + b)\). We are given that the perimeter is \(80\), so \(2 (l + b) = 80\), which simplifies to \(l + b = 40\). From this, we can express the breadth as \(b = 40 − l\).

The area of the park is given by \(l \times b\). Substituting \(b = 40 - l\), the area becomes \(l (40 - l)\), which expands to \(40l - l^2\). We are given that the area is \(400\), so \(40l - l^2 = 400\). Rearranging this equation gives \(l^2 - 40l + 400 = 0\).

Comparing this quadratic equation with the standard form \(al^2 + bl + c = 0\), we identify the coefficients: \(a = 1\), \(b = -40\), and \(c = 400\).

The discriminant of the quadratic equation is calculated as \(b^2 - 4ac\). Substituting the values, we get \((-40)^2 - 4(1)(400) = 1600 - 1600 = 0\). Since the discriminant \(b^2 - 4ac = 0\), the equation has equal real roots, indicating that this geometric situation is possible.

The roots of this quadratic equation are given by \(l = -\frac{b}{2a}\).

Substituting the values of \(a\) and \(b\), we find \(l = -\frac{(-40)}{2(1)} = \frac{40}{2} = 20\).

Therefore, the length of the park is \(l = 20\) m. The breadth of the park is \(b = 40 - l = 40 - 20 = 20\) m.