To determine who will raise [tex]$100 with the fewest number of laps, we need to analyze the rate of change, or slope, for each person's data.
### Step 1: Calculating the Slope
The slope \( m \) of a line can be determined using two points \((x_1, y_1)\) and \((x_2, y_2)\) with the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
#### Michelle's Data:
Michelle's data points are \((10, 15)\), \((30, 45)\), and \((60, 90)\).
Using points \((10, 15)\) and \((60, 90)\):
\[ m = \frac{90 - 15}{60 - 10} = \frac{75}{50} = 1.5 \]
So, the slope for Michelle is 1.5.
#### Andrea's Data:
Andrea's data points are \((20, 25)\), \((32, 40)\), and \((40, 50)\).
Using points \((20, 25)\) and \((40, 50)\):
\[ m = \frac{50 - 25}{40 - 20} = \frac{25}{20} = 1.25 \]
So, the slope for Andrea is 1.25.
### Step 2: Calculate the Number of Laps Needed to Raise $[/tex]100
If the slope represents the amount of money raised per lap, we can determine the required number of laps to raise [tex]$100 by dividing $[/tex]100 by the slope.
#### Michelle:
[tex]\[ \text{Laps for Michelle} = \frac{100}{1.5} \approx 66.67 \text{ laps} \][/tex]
#### Andrea:
[tex]\[ \text{Laps for Andrea} = \frac{100}{1.25} = 80 \text{ laps} \][/tex]
### Step 3: Determine Who Will Raise [tex]$100 with the Fewest Number of Laps
From the calculations above:
- Michelle needs approximately 66.67 laps.
- Andrea needs 80 laps.
Since 66.67 laps are fewer than 80 laps, Michelle will be able to raise $[/tex]100 with the fewest number of laps.
### Conclusion
The correct statement is:
Michelle will, because the slope of the line described by the data in her table is the greatest.
