NCERT Class X Chapter 4: Quadratic Equation Exercise 4.3 Question 3

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.3 Question 3

Question:

Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m^2? If so, find its length and breadth.

Given:

A rectangular mango grove of

Length (L) is twice its breadth (B) ⇒ L = 2B

Area (A) = 800 m2

To Find:

1. Is it possible to design such a rectangular mango grove?

2. If yes, find its length and breadth.

Formula:

Area of a rectangle is given by: Area = Length × Breadth (A = L × B)

A quadratic equation of the form ax2 + bx + c = 0, for real solutions to exist, the discriminant (Δ) must be greater than or equal to zero.

Δ = b2 - 4ac ≥ 0

Solution:

Let the breadth of the rectangular mango grove be 'x' meters.

According to the problem, the length is twice its breadth:

Length = 2x meters.
The area of the rectangular mango grove is given as 800 m2.

WKT, Area = Length × Breadth

⇒ 800 = (2x) × (x)

⇒ 800 = 2x2
Group terms on one side of '=' sign
⇒ 2x2 - 800 = 0

Take 2 from each term since it is common
⇒ 2(x2 - 400) = 0

Divide the entire equation by 2 to simplify:
⇒ x2 - 400 = 0
Comparing x2 - 400 = 0 with ax2 + bx + c = 0, we have:
a = 1, b = 0, and c = -400
To determine if it's possible to design such a grove, check the discriminant (Δ):

WKT, Δ = b2 - 4ac

⇒ Δ = (0)2 - 4(1)(-400)

⇒ Δ = 0 + 1600

⇒ Δ = 1600

Since Δ = 1600 > 0, real roots exist, which means it is possible to design such a mango grove.
Now, find the values of x (breadth) by solving x2 - 400 = 0:

⇒ x2 = 400

⇒ x = ±√400

⇒ x = ±20

Since breadth cannot be negative, we take the positive value:

Breadth (x) = 20 meters.


Calculate the length:

Length = 2x = 2 × 20 = 40 meters.

Result:

Yes, it is possible to design a rectangular mango grove with the given conditions.

The length of the mango grove is 40 meters, and its breadth is 20 meters.
© Kaliyuga Ekalavya. All rights reserved.

Comments

Post a Comment