To derive an equation that connects the dimensions from the scale drawing to the actual dimensions of the rectangular playing field, we start with the given scale: 2 inches on the drawing corresponds to 3 feet in reality.
We can find the conversion factor by setting up a proportion based on the scale:
\[ \frac{\text{actual feet}}{\text{drawing inches}} = \frac{3 \text{ feet}}{2 \text{ inches}}. \]
To find the relationship between the dimensions of the scale drawing (in inches) and the actual dimensions of the field (in feet), we can express it in the form \(y = mx\), where:
- \(y\) is the actual dimension in feet,
- \(x\) is the dimension in inches,
- \(m\) is the conversion factor.
From the scale, we can see that:
\[ m = \frac{3 \text{ feet}}{2 \text{ inches}} = 1.5 \text{ feet per inch}. \]
Thus, the equation that relates the dimension in inches to the actual dimension in feet is:
\[ y = 1.5x. \]
Next, we can use this equation to find the actual dimensions of the playing field given its dimensions from the scale drawing:
- Length from the drawing: \(12\) inches
- Width from the drawing: \(6\) inches
Now we can calculate the actual length and width using the equation \(y = 1.5x\).
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Actual Length: \[ y = 1.5 \times 12 = 18 \text{ feet}. \]
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Actual Width: \[ y = 1.5 \times 6 = 9 \text{ feet}. \]
Final Answer:
- The equation is \(y = 1.5x\).
- Actual Length: 18 feet.
- Actual Width: 9 feet.
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