Sure! Let's solve this step by step.
## Part (a)
### Given:
- The area of the rectangular playground is 300 m².
- The length of the playground [tex]\( l \)[/tex] is 1 meter more than twice its breadth [tex]\( b \)[/tex].
This translates to:
[tex]\[ l = 2b + 1 \][/tex]
[tex]\[ \text{Area} = l \times b = 300 \][/tex]
### Step-by-step solution:
1. Substitute the expression for [tex]\( l \)[/tex] into the area equation:
[tex]\[ (2b + 1) \times b = 300 \][/tex]
2. Simplify and solve the quadratic equation:
[tex]\[ 2b^2 + b - 300 = 0 \][/tex]
3. Solve the quadratic equation using factoring, the quadratic formula, or another method:
[tex]\[ b = 12 \][/tex]
4. Substitute the value of [tex]\( b \)[/tex] back into the length equation to find [tex]\( l \)[/tex]:
[tex]\[ l = 2(12) + 1 \][/tex]
[tex]\[ l = 25 \][/tex]
### Answer for Part (a):
- Breadth of the playground [tex]\( b = 12 \)[/tex] meters
- Length of the playground [tex]\( l = 25 \)[/tex] meters
## Part (b)
### Given:
- We want to convert the rectangular playground (keeping the area the same) to a square.
### Step-by-step solution:
1. Find the side length of the square. Since the area of the playground is to remain 300 m², the side length [tex]\( s \)[/tex] of the square is:
[tex]\[ s = \sqrt{300} \][/tex]
[tex]\[ s \approx 17.32 \][/tex]
2. Calculate the decrease needed in the length:
[tex]\[ \text{Length decrease} = l - s \][/tex]
[tex]\[ \text{Length decrease} = 25 - 17.32 \][/tex]
[tex]\[ \text{Length decrease} \approx 7.68 \][/tex]
3. Determine the percentage decrease in length:
[tex]\[ \text{Percentage decrease} = \left( \frac{\text{Length decrease}}{l} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage decrease} = \left( \frac{7.68}{25} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage decrease} \approx 30.72 \% \][/tex]
### Answer for Part (b):
- The length of the playground should be decreased by approximately 30.72% to convert it to a square with the same area.
Thus, these detailed steps lead to the results:
- Breadth of the playground: 12 meters
- Length of the playground: 25 meters
- Side length of the equivalent square: 17.32 meters
- Percentage decrease in length: approximately 30.72%
