Question

angela is training for a marathon and completes her long mileage runs for training on the weekend. over the last 3 weekends she ran 15 miles in 2 hours; 18 miles in 2 hours 33 minutes; and 22 miles in 3 hours 7 minutes. determine if her weekend training runs showcase a proportional relationship

Answer 1

To determine if Angela's weekend training runs showcase a proportional relationship, we need to check if the ratio of distance to time for each of her runs is constant.

A proportional relationship means that if we take the distance divided by the time for each run (miles per hour), the results should be the same. Let's calculate the ratios for each of her long runs.

1. **First run:**
Distance: 15 miles
Time: 2 hours = 2 hours * 60 minutes/hour = 120 minutes
Ratio = \( \frac{15 \text{ miles}}{2 \text{ hours}} = \frac{15}{2} = 7.5 \) miles per hour

2. **Second run:**
Distance: 18 miles
Time: 2 hours 33 minutes = 2 * 60 + 33 = 153 minutes = 153/60 hours = 2.55 hours
Ratio = \( \frac{18 \text{ miles}}{2.55 \text{ hours}} \approx 7.06 \) miles per hour

3. **Third run:**
Distance: 22 miles
Time: 3 hours 7 minutes = 3 * 60 + 7 = 187 minutes = 187/60 hours \approx 3.12 hours
Ratio = \( \frac{22 \text{ miles}}{3.12 \text{ hours}} \approx 7.05 \) miles per hour

Now, let's summarize the results:
- First run: 7.5 miles/hour
- Second run: approximately 7.06 miles/hour
- Third run: approximately 7.05 miles/hour

Since the ratios (miles per hour) for each of her long runs are not the same, Angela's weekend training runs do not showcase a proportional relationship.