1. The length of a playground is [tex]$7 \, \text{m}$[/tex] less than double its breadth. The area of the land is [tex]$400 \, \text{m}^2$[/tex].

a. Write the above statement in the form of a quadratic equation.

b. Find the length and breadth of the ground.

c. By what percentage should the length be reduced to make it a square?



Answer :

Certainly! Let’s go through the solution step by step.

### Part a: Formulating the quadratic equation

1. Let the breadth of the playground be [tex]\( b \)[/tex] meters.

2. Express the length of the playground:
The problem states that the length is 7 meters less than double the breadth. So, the length [tex]\( l \)[/tex] can be expressed as:
[tex]\[ l = 2b - 7 \][/tex]

3. Write the relationship between the area, length, and breadth:
The area [tex]\( A \)[/tex] of a rectangle is given by the product of its length and its breadth. According to the problem, the area is 400 square meters:
[tex]\[ A = l \cdot b \\ 400 = (2b - 7) \cdot b \][/tex]

4. Formulate the quadratic equation:
Expanding the equation, we get:
[tex]\[ 400 = 2b^2 - 7b \][/tex]
Bringing all terms to one side of the equation, we have:
[tex]\[ 2b^2 - 7b - 400 = 0 \][/tex]
This is our quadratic equation.

### Part b: Finding the Length and Breadth

1. Solving the quadratic equation:
The quadratic equation is:
[tex]\[ 2b^2 - 7b - 400 = 0 \][/tex]
Solving this quadratic equation for [tex]\( b \)[/tex], we get two solutions. However, we are only interested in the positive real solutions (as measures must be positive):
[tex]\[ b = 16 \][/tex]

2. Finding the corresponding length:
Substitute the value of [tex]\( b \)[/tex] back into the expression for the length:
[tex]\[ l = 2b - 7 \\ l = 2 \cdot 16 - 7 \\ l = 32 - 7 \\ l = 25 \][/tex]

So, the breadth of the ground is [tex]\( 16 \)[/tex] meters, and the length is [tex]\( 25 \)[/tex] meters.

### Part c: Percentage Reduction to Make the Ground a Square

1. Current dimensions:
- Length, [tex]\( l = 25 \)[/tex] meters
- Breadth, [tex]\( b = 16 \)[/tex] meters

2. To make the playground a square:
For the figure to be a square, the length should equal the breadth. Therefore, the length should be reduced to [tex]\( 16 \)[/tex] meters.

3. Calculating the percentage reduction:
The original length is [tex]\( 25 \)[/tex] meters, and it needs to be reduced to [tex]\( 16 \)[/tex] meters. The reduction in length is:
[tex]\[ \text{Reduction} = 25 - 16 = 9 \text{ meters} \][/tex]
The percentage reduction is then:
[tex]\[ \text{Percentage Reduction} = \left( \frac{\text{Reduction}}{\text{Original Length}} \right) \times 100 \\ \text{Percentage Reduction} = \left( \frac{9}{25} \right) \times 100 \\ \text{Percentage Reduction} = 36\% \][/tex]

### Summary

a) The quadratic equation you get is:
[tex]\[ 2b^2 - 7b - 400 = 0 \][/tex]

b) The breadth of the ground is [tex]\( 16 \)[/tex] meters and the length is [tex]\( 25 \)[/tex] meters.

c) The length should be reduced by [tex]\( 36\% \)[/tex] to make the playground a square.