NCERT Class X Chapter 4: Quadratic Equation Example 6

NCERT Class X Chapter 4: Quadratic Equation Example 6

Question:

Charity trust decides to build a prayer hall having a carpet area of 300 square metres with its length one metre more than twice its breadth. What should be the length and breadth of the hall?.

Given:

Carpet area of the prayer hall = 300 m2

Length = 1 metre more than twice its breadth.

To Find:

The length of the hall.

The breadth of the hall

Formula:

Area of a rectangle = Length × Breadth.
For a quadratic equation ax2 + bx + c = 0, the roots are given by the quadratic formula:

x = -b ± √(Δ) 2a

Δ = b2 - 4ac

Solution:

Let the breadth of the hall (in metres) = x
According to the given condition, the length is one metre more than twice its breadth.

⇒ Length (in metres) = ( 1 + 2x ) = ( 2x + 1 )
Area of the hall = Length × Breadth.

⇒ 300 = (2x + 1) × x

⇒ 300 = 2x2 + x
Group the terms on one side of the '=' sign

⇒ 2x2 + x - 300 = 0 ( A quadratic equation )
Now, solve this quadratic equation using the quadratic formula.

Here, a = 2, b = 1, c = -300.

Calculate the discriminant (Δ = b2 - 4ac):

⇒ Δ = (1)2 - 4(2)(-300)

⇒ Δ = 1 + 2400

⇒ Δ = 2401
Now, calculate √Δ:

⇒ √Δ = √2401 = 49

Substitute the values into the quadratic formula: x = -b ± √(Δ) 2a

x = -1 ± 49 2 × 2


x = -1 ± 49 4
Calculate the two possible values for x:

x1 = -1 + 49 4 = 48 4 = 12

x2 = -1 - 49 4 = -50 4 = -12.5
Since breadth cannot be negative, we take x = 12 metres.

Therefore, Breadth = 12 metres.

Substitute x = 12 in 2x+1 to find the Length.

⇒ Length = 2x + 1

⇒ Length = 2(12) + 1

⇒ Length = 24 + 1

⇒ Length = 25 metres.

Result:

The breadth of the hall is 12 metres and the length of the hall is 25 metres.
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