Question

Find length and breadth of a rectangular park whose perimeter is \(100 \, \text{m}\) and area is \(600 \, \text{m}^2\).

Hint:

Use perimeter formula to express one variable in terms of the other and substitute into area equation.

Answer 1

Given:
Perimeter of rectangular park, \(P = 100 \, \text{m}\)
Area of rectangular park, \(A = 600 \, \text{m}^2\)
Let length = \(l\) m and breadth = \(b\) m

Step 1: Write formulas for perimeter and area
Perimeter: \(P = 2(l + b)\)
Area: \(A = l \times b\)

Step 2: Use the perimeter formula to find relation between \(l\) and \(b\)
\[ 2(l + b) = 100 \implies l + b = 50 \implies l = 50 - b \]

Step 3: Use the area formula
\[ l \times b = 600 \]
Substitute \(l = 50 - b\):
\[ (50 - b) \times b = 600 \Rightarrow 50b - b^2 = 600 \Rightarrow b^2 - 50b + 600 = 0 \]

Step 4: Solve quadratic equation \(b^2 - 50b + 600 = 0\)
Calculate discriminant \(D\):
\[ D = (-50)^2 - 4 \times 1 \times 600 = 2500 - 2400 = 100 \]
\[ b = \frac{50 \pm \sqrt{100}}{2} = \frac{50 \pm 10}{2} \]
So,
\[ b = \frac{50 + 10}{2} = \frac{60}{2} = 30 \quad \text{or} \quad b = \frac{50 - 10}{2} = \frac{40}{2} = 20 \]

Step 5: Find corresponding length \(l\)
If \(b = 30\), then \(l = 50 - 30 = 20\)
If \(b = 20\), then \(l = 50 - 20 = 30\)

Final Answer:
Length = 30 m, Breadth = 20 m
(or Length = 20 m, Breadth = 30 m)