To determine who will raise $100 with the fewest number of laps, we first need to analyze the data provided for both Michelle and Andrea to determine the slope (rate of money raised per lap) for each.
Michelle's Data:
- (10 laps, $15)
- (30 laps, $45)
- (60 laps, $90)
To find the slope for Michelle:
- For the first two data points: \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{45 - 15}{30 - 10} = \frac{30}{20} = 1.5 \]
- For the second two data points: \[ \text{slope} = \frac{90 - 45}{60 - 30} = \frac{45}{30} = 1.5 \]
Michelle's slope is consistently 1.5 dollars per lap.
Andrea's Data:
- (20 laps, $25)
- (32 laps, $40)
- (40 laps, $50)
To find the slope for Andrea:
- For the first two data points: \[ \text{slope} = \frac{40 - 25}{32 - 20} = \frac{15}{12} = 1.25 \]
- For the second two data points: \[ \text{slope} = \frac{50 - 40}{40 - 32} = \frac{10}{8} = 1.25 \]
Andrea's slope is consistently 1.25 dollars per lap.
Comparison:
- Michelle raises $1.5 per lap.
- Andrea raises $1.25 per lap.
To find out how many laps each girl needs to complete to raise $100, we can use the formula:
\[ \text{laps required} = \frac{\text{desired amount}}{\text{slope}} \]
For Michelle: \[ \text{laps required} = \frac{100}{1.5} \approx 66.67 \text{ laps} \]
For Andrea: \[ \text{laps required} = \frac{100}{1.25} = 80 \text{ laps} \]
Conclusion: Michelle will raise $100 with approximately 67 laps, while Andrea will need 80 laps.
The correct statement is: Michelle will, because the slope of the line described by the data in her table is the greatest.
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