Find the dimensions of a rectangular park whose perimeter is 60 m and area 200 `m^(2)`

Answer

To find the dimensions of a rectangular park whose perimeter is 60 m and area is 200 m², we can follow these steps:

### Step 1: Set up the equations
Let the length of the park be \( L \) meters and the breadth be \( B \) meters.

From the problem, we know:
1. The perimeter of the rectangle is given by the formula:
\[ P = 2(L + B) = 60 \]
Dividing both sides by 2, we get:
\[ L + B = 30 \quad \text{(Equation 1)} \]

2. The area of the rectangle is given by the formula:
\[ A = L \times B = 200 \quad \text{(Equation 2)} \]

### Step 2: Express one variable in terms of the other
From Equation 1, we can express \( B \) in terms of \( L \):
\[ B = 30 - L \quad \text{(Equation 3)} \]

### Step 3: Substitute Equation 3 into Equation 2
Now, substitute Equation 3 into Equation 2:
\[ L \times (30 - L) = 200 \]
Expanding this gives:
\[ 30L - L^2 = 200 \]
Rearranging the equation, we get:
\[ L^2 - 30L + 200 = 0 \quad \text{(Equation 4)} \]

### Step 4: Solve the quadratic equation
Now we will solve the quadratic equation \( L^2 - 30L + 200 = 0 \) using the factorization method.

We need to find two numbers that multiply to \( 200 \) and add up to \( 30 \). The numbers are \( 20 \) and \( 10 \):
\[ L^2 - 20L - 10L + 200 = 0 \]
Grouping the terms:
\[ L(L - 20) - 10(L - 20) = 0 \]
Factoring out \( (L - 20) \):
\[ (L - 20)(L - 10) = 0 \]

### Step 5: Find the values of \( L \)
Setting each factor to zero gives us:
1. \( L - 20 = 0 \) → \( L = 20 \)
2. \( L - 10 = 0 \) → \( L = 10 \)

### Step 6: Find the corresponding values of \( B \)
Now we can find the corresponding values of \( B \) using Equation 3:
- If \( L = 20 \):
\[ B = 30 - 20 = 10 \]
- If \( L = 10 \):
\[ B = 30 - 10 = 20 \]

### Conclusion
Thus, the dimensions of the rectangular park are:
- Length \( L = 20 \) m and Breadth \( B = 10 \) m
or
- Length \( L = 10 \) m and Breadth \( B = 20 \) m