Answer :
Let's solve the problem step by step.
1. Define the Variables:
- Let [tex]\( l \)[/tex] be the length of the rectangle.
- The breadth of the rectangle is [tex]\( b = l - 5 \)[/tex] meters.
2. Area of the Rectangle:
- The area of the rectangle is given as 54 square meters.
- Therefore, the equation representing the area is:
[tex]\[ l \times (l - 5) = 54 \][/tex]
3. Formulate and Solve the Quadratic Equation:
- Expanding the area equation:
[tex]\[ l^2 - 5l = 54 \][/tex]
- Rearrange it into standard quadratic form:
[tex]\[ l^2 - 5l - 54 = 0 \][/tex]
- Solving this quadratic equation, we get two solutions:
[tex]\[ l = \frac{5}{2} - \frac{\sqrt{241}}{2} \quad \text{(negative, not valid)} \\ l = \frac{5}{2} + \frac{\sqrt{241}}{2} \quad \text{(positive, valid)} \][/tex]
4. Calculate the Breadth:
- Substitute the valid solution for the length back to find the breadth:
[tex]\[ b = \left( \frac{5}{2} + \frac{\sqrt{241}}{2} \right) - 5 \][/tex]
5. Calculate the Perimeter:
- The perimeter [tex]\( P \)[/tex] of a rectangle is given by:
[tex]\[ P = 2 \times (l + b) \][/tex]
- Substituting the values of [tex]\( l \)[/tex] and [tex]\( b \)[/tex] into the perimeter equation, we get:
[tex]\[ P = 2 \times \left( \left( \frac{5}{2} + \frac{\sqrt{241}}{2} \right) + \left( \frac{5}{2} + \frac{\sqrt{241}}{2} - 5 \right) \right) \][/tex]
- Simplifying the expression inside the parentheses:
[tex]\[ l + b = \left( \frac{5}{2} + \frac{\sqrt{241}}{2} \right) + \left( \frac{5}{2} + \frac{\sqrt{241}}{2} - 5 \right) = \sqrt{241} \][/tex]
- Therefore, the perimeter is:
[tex]\[ P = 2 \times \sqrt{241} \][/tex]
6. Final Answer:
The correct answer, from the given options, is:
[tex]\[ \boxed{2 \times \sqrt{241} \text{ meters}} \][/tex]
However, the answer option choices given (a, b, c, d) don't seem to correspond to any valid perimeter for this problem based on typical units used. It seems there may be an error in the choices provided, but the calculated perimeter: [tex]\( \boxed{2 \times \sqrt{241} \text{ meters}} \)[/tex].
Since [tex]\(2 \times \sqrt{241} \approx 31 \text{ meters} \)[/tex], it is best to reconsider the validity of the provided options. The closest logical step would be to verify the correct match from valid options or recheck area or units.
1. Define the Variables:
- Let [tex]\( l \)[/tex] be the length of the rectangle.
- The breadth of the rectangle is [tex]\( b = l - 5 \)[/tex] meters.
2. Area of the Rectangle:
- The area of the rectangle is given as 54 square meters.
- Therefore, the equation representing the area is:
[tex]\[ l \times (l - 5) = 54 \][/tex]
3. Formulate and Solve the Quadratic Equation:
- Expanding the area equation:
[tex]\[ l^2 - 5l = 54 \][/tex]
- Rearrange it into standard quadratic form:
[tex]\[ l^2 - 5l - 54 = 0 \][/tex]
- Solving this quadratic equation, we get two solutions:
[tex]\[ l = \frac{5}{2} - \frac{\sqrt{241}}{2} \quad \text{(negative, not valid)} \\ l = \frac{5}{2} + \frac{\sqrt{241}}{2} \quad \text{(positive, valid)} \][/tex]
4. Calculate the Breadth:
- Substitute the valid solution for the length back to find the breadth:
[tex]\[ b = \left( \frac{5}{2} + \frac{\sqrt{241}}{2} \right) - 5 \][/tex]
5. Calculate the Perimeter:
- The perimeter [tex]\( P \)[/tex] of a rectangle is given by:
[tex]\[ P = 2 \times (l + b) \][/tex]
- Substituting the values of [tex]\( l \)[/tex] and [tex]\( b \)[/tex] into the perimeter equation, we get:
[tex]\[ P = 2 \times \left( \left( \frac{5}{2} + \frac{\sqrt{241}}{2} \right) + \left( \frac{5}{2} + \frac{\sqrt{241}}{2} - 5 \right) \right) \][/tex]
- Simplifying the expression inside the parentheses:
[tex]\[ l + b = \left( \frac{5}{2} + \frac{\sqrt{241}}{2} \right) + \left( \frac{5}{2} + \frac{\sqrt{241}}{2} - 5 \right) = \sqrt{241} \][/tex]
- Therefore, the perimeter is:
[tex]\[ P = 2 \times \sqrt{241} \][/tex]
6. Final Answer:
The correct answer, from the given options, is:
[tex]\[ \boxed{2 \times \sqrt{241} \text{ meters}} \][/tex]
However, the answer option choices given (a, b, c, d) don't seem to correspond to any valid perimeter for this problem based on typical units used. It seems there may be an error in the choices provided, but the calculated perimeter: [tex]\( \boxed{2 \times \sqrt{241} \text{ meters}} \)[/tex].
Since [tex]\(2 \times \sqrt{241} \approx 31 \text{ meters} \)[/tex], it is best to reconsider the validity of the provided options. The closest logical step would be to verify the correct match from valid options or recheck area or units.
