NCERT Class X Chapter 4: Quadratic Equation Exercise 4.3 Question 5
NCERT Class X Chapter 4: Quadratic Equation Exercise 4.3 Question 5
Question:
Is it possible to design a rectangular park of perimeter 80 m and area 400 m^2? If so, find its length and breadth.
Given:
A rectangular park withPerimeter (P) = 80 m
Area (A) = 400 m2
To Find:
1. Is it possible to design such a rectangular park?2. If yes, find its length and breadth.
Formula:
- Perimeter of a rectangle is given by: P = 2(Length + Breadth)
- Area of a rectangle is given by: A = Length × Breadth
-
For a quadratic equation of the form ax2 + bx + c = 0, real solutions exist if and only if its discriminant (Δ) is greater than or equal to zero.
Δ = b2 - 4ac ≥ 0
Solution:
Let the length of the rectangular park be 'l' meters
Let the breadth of the rectangular park be 'b' meters.
Let the breadth of the rectangular park be 'b' meters.
Given Perimeter (P) = 80 m
WKT, P = 2(l + b)
⇒ 2(l + b) = 80
⇒ l + b = 40 (Equation 1)
WKT, P = 2(l + b)
⇒ 2(l + b) = 80
⇒ l + b = 40 (Equation 1)
From Equation 1, we can express length in terms of breadth:
⇒ l = 40 - b
⇒ l = 40 - b
Given Area (A) = 400 m2
WKT, A = l × b
⇒ 400 = (40 - b) × b
⇒ 400 = 40b - b2
Grouping the terms on one side of '=' we have
b2 - 40b + 400 = 0
WKT, A = l × b
⇒ 400 = (40 - b) × b
⇒ 400 = 40b - b2
Grouping the terms on one side of '=' we have
b2 - 40b + 400 = 0
Comparing b2 - 40b + 400 = 0 this with ax2 + bx + c = 0, we have:
a = 1
b = -40
c = 400
a = 1
b = -40
c = 400
To determine if it's possible to design such a park, we calculate the discriminant (Δ):
WKT, Δ = b2 - 4ac
⇒ Δ = (-40)2 - 4(1)(400)
⇒ Δ = 1600 - 1600
⇒ Δ = 0
Since Δ = 0, the quadratic equation has two equal real roots it is possible to design such a park.
WKT, Δ = b2 - 4ac
⇒ Δ = (-40)2 - 4(1)(400)
⇒ Δ = 1600 - 1600
⇒ Δ = 0
Since Δ = 0, the quadratic equation has two equal real roots it is possible to design such a park.
Now,
Lets find the values of 'b' (breadth) by solving the quadratic equation b2 - 40b + 400 = 0
b2 - 40b + 400 = 0 can be written as b2 - 2(1)(20)b + (20)(20) = 0
This is of the form a2 - 2ab + b2 which is equal to (a - b)2
Here a = b and b = 20;
We can factor this as a perfect square:
⇒ (b - 20)2 = 0
⇒ b - 20 = 0
⇒ b = 20
So, Breadth (b) = 20 meters.
Find the length 'l' using Equation 1 (l = 40 - b):
⇒ l = 40 - 20
⇒ l = 20 meters.
Lets find the values of 'b' (breadth) by solving the quadratic equation b2 - 40b + 400 = 0
b2 - 40b + 400 = 0 can be written as b2 - 2(1)(20)b + (20)(20) = 0
This is of the form a2 - 2ab + b2 which is equal to (a - b)2
Here a = b and b = 20;
We can factor this as a perfect square:
⇒ (b - 20)2 = 0
⇒ b - 20 = 0
⇒ b = 20
So, Breadth (b) = 20 meters.
Find the length 'l' using Equation 1 (l = 40 - b):
⇒ l = 40 - 20
⇒ l = 20 meters.
Result:
Yes, it is possible to design a rectangular park with the given conditions.The length of the park is 20 meters, and its breadth is 20 meters.
Since the length and breadth of the park is equal it is would be a square park.
Next Question Solution:
NCERT Class X Chapter 5: Arithmetic Progressions Example 1.Explore more in Quadratic Equations chapter:
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