Is it possible to design a rectangular park of perimeter 80 m and area 400 m2? If so, find its length and breadth.
Step-by-Step Solution
Key Concept: Use the relations between perimeter and area of a rectangle. Express one side in terms of the other using the perimeter condition, substitute into the area condition, and obtain a quadratic equation. Solve the quadratic using the discriminant method taught in NCERT.
1. Let the length be \(l\) metres and the breadth be \(b\) metres.
2. Perimeter condition:
$$2(l+b)=80 \;\Rightarrow\; l+b=40 \quad\text{(Equation 1)}$$
3. Area condition:
$$l\times b = 400 \quad\text{(Equation 2)}$$
4. Express \(b\) from Equation 1:
$$b = 40 - l$$
5. Substitute \(b\) in Equation 2:
$$l(40 - l) = 400$$
6. Form the quadratic equation:
$$40l - l^{2} = 400 \;\Rightarrow\; l^{2} - 40l + 400 = 0$$
7. Identify coefficients:
$$a = 1,\; b = -40,\; c = 400$$
8. Compute the discriminant \(D\):
$$D = b^{2} - 4ac = (-40)^{2} - 4\times1\times400 = 1600 - 1600 = 0$$
9. Since \(D = 0\), the quadratic has equal real roots:
$$l = \frac{-b}{2a} = \frac{40}{2} = 20\;\text{metres}$$
10. Find \(b\) using Equation 1:
$$b = 40 - l = 40 - 20 = 20\;\text{metres}$$
11. Conclusion: Both length and breadth are 20 m, i.e., the rectangle is a square. Hence, such a park can be designed.
Answer: Length = 20 m, Breadth = 20 m.
Correct Answer: Yes, it is possible. Length = 20 m and Breadth = 20 m.