To derive the equation of the proportional relationship through the origin, we first need to determine the relationship between the cost \( y \) and the quantity \( x \) of footballs.
From the data provided:
- Dot 2: \( (1.9, 2) \)
- Dot 3: \( (3.27, 3) \)
- Dot 4: \( (4.36, 4) \)
- Dot 5: \( (5.48, 5) \)
To find the equation of a proportional relationship in the form \( y = kx \), where \( k \) is the constant of proportionality, we can calculate \( k \) for each point using the formula:
\[ k = \frac{y}{x} \]
Calculating \( k \) for each dot:
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For \( (2, 1.9) \): \[ k = \frac{1.9}{2} = 0.95 \]
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For \( (3, 3.27) \): \[ k = \frac{3.27}{3} = 1.09 \]
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For \( (4, 4.36) \): \[ k = \frac{4.36}{4} = 1.09 \]
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For \( (5, 5.48) \): \[ k = \frac{5.48}{5} = 1.096 \]
It seems that the value of \( k \) is not consistent across all points, which implies the relationship is not perfectly proportional. However, we can use the average of the calculated \( k \) values to find an approximate relationship.
Calculating the average \( k \): \[ k \approx \frac{0.95 + 1.09 + 1.09 + 1.096}{4} = \frac{4.233}{4} \approx 1.05825 \]
Thus, we can generally express the relationship using the equation: \[ y \approx 1.06x \]
This equation represents the approximate linear relationship between the cost and the quantity of footballs, assuming a proportional relationship through the origin.