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

To determine if it is possible to design a rectangular park with a perimeter of 80 m and an area of 400 m², we can use the formulas for perimeter and area of a rectangle.

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

1. The formula for the perimeter \( P \) of a rectangle is given by:
\[ P = 2(L + B) \]
Given that the perimeter is 80 m, we can write:
\[ 2(L + B) = 80 \]
Dividing both sides by 2, we get:
\[ L + B = 40 \quad \text{(Equation 1)} \]

2. The formula for the area \( A \) of a rectangle is given by:
\[ A = L \times B \]
Given that the area is 400 m², we can write:
\[ L \times B = 400 \quad \text{(Equation 2)} \]

### Step 2: Express one variable in terms of the other
From Equation 1, we can express \( B \) in terms of \( L \):
\[ B = 40 - L \quad \text{(Substituting into Equation 2)} \]

### Step 3: Substitute into the area equation
Now substitute \( B \) in Equation 2:
\[ L \times (40 - L) = 400 \]
Expanding this, we have:
\[ 40L - L^2 = 400 \]

### Step 4: Rearrange into standard quadratic form
Rearranging gives us:
\[ -L^2 + 40L - 400 = 0 \]
Multiplying through by -1 to make the leading coefficient positive:
\[ L^2 - 40L + 400 = 0 \]

### Step 5: Solve the quadratic equation
Now we can solve this quadratic equation using the quadratic formula:
\[ L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -40, c = 400 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-40)^2 - 4 \times 1 \times 400 = 1600 - 1600 = 0 \]

Since the discriminant is 0, there is one real solution:
\[ L = \frac{40 \pm \sqrt{0}}{2 \times 1} = \frac{40}{2} = 20 \]

### Step 6: Find the breadth
Now that we have \( L = 20 \) m, we can find \( B \):
\[ B = 40 - L = 40 - 20 = 20 \text{ m} \]

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

### Final Answer
Yes, it is possible to design the park with the given dimensions.

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