To determine if the area of a square is a function of its perimeter, we first need to analyze the relationships between the perimeter and the area using the formulas for a square.
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Formulas:
- The perimeter \( P \) of a square with side length \( s \) is given by: \[ P = 4s \]
- The area \( A \) of a square is given by: \[ A = s^2 \]
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Relating Area to Perimeter: From the perimeter formula, we can express the side length \( s \) in terms of the perimeter: \[ s = \frac{P}{4} \] Substituting this into the area formula, we get: \[ A = s^2 = \left(\frac{P}{4}\right)^2 = \frac{P^2}{16} \] This shows that the area can be expressed as a function of the perimeter.
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Verifying the Given Data: Let's check the provided values:
- For \( P = 12 \): \[ s = \frac{12}{4} = 3 \quad \Rightarrow \quad A = 3^2 = 9 \]
- For \( P = 20 \): \[ s = \frac{20}{4} = 5 \quad \Rightarrow \quad A = 5^2 = 25 \]
- For \( P = 36 \): \[ s = \frac{36}{4} = 9 \quad \Rightarrow \quad A = 9^2 = 81 \]
All calculations match the provided data.
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Conclusion: Based on the analysis, the area of a square is indeed a function of its perimeter. For every input (perimeter), there is exactly one output (area). This means that: The area of a square is a function of its perimeter because for every input there is exactly one possible output.
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