Find length and breadth of a rectangular park whose perimeter is $100$ m and area is $600$ $m^2$

Find length and breadth of a rectangular park whose perimeter is $100$ m and area is $600$ $m^2$

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

### Step 1: Use the Perimeter Equation: The perimeter of a rectangle is given by:

$ 2(l + b) = 100 $

Dividing by 2:

$ l + b = 50 $ ----(1)

### Step 2: Use the Area Equation: The area of a rectangle is given by:

$ l \times b = 600 $ ----(2)

### Step 3: Solve for $ l $ and $ b $: From equation (1), express $ l $ in terms of $ b $:

$ l = 50 - b $

Substituting in equation (2):

$ (50 - b) \times b = 600 $

Expanding:

$ 50b - b^2 = 600 $

Rearranging:

$ b^2 - 50b + 600 = 0 $

### Step 4: Solve the Quadratic Equation: Using the quadratic formula where:

$ a = 1, \quad b = -50, \quad c = 600 $

The roots are given by:

$ b = \frac{-(-50) \pm \sqrt{(-50)^2 - 4(1)(600)}}{2(1)} $

$ b = \frac{50 \pm \sqrt{2500 - 2400}}{2} $

$ b = \frac{50 \pm \sqrt{100}}{2} $

$ b = \frac{50 \pm 10}{2} $

Solving for $ b $:

$ b = \frac{50 + 10}{2} = \frac{60}{2} = 30 $

or

$ b = \frac{50 - 10}{2} = \frac{40}{2} = 20 $

Thus, the possible dimensions are:
- If $ b = 30 $, then $ l = 20 $.
- If $ b = 20 $, then $ l = 30 $.

### Final Answer: The length and breadth of the rectangular park are $ 30 $ m and $ 20 $ m (or vice versa).

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