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Sagot :
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.
### 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.
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