Question

The table given below shows the number of miles that Robert has been running over some of the past 8 weeks:1 week: 5 miles, 2 weeks: 6.5 miles, 3 weeks: 8 miles, 5 weeks: 11 miles, and 8 weeks: 15.5 miles.The number of miles that he runs per week can be represented as a linear function. What would be the rate of change of this function in miles per week?

Answer 1

The rate of change of this function, which represents the number of miles Robert runs per week, is 1.5 miles per week.

To find the rate of change, which represents the slope of the linear function in the context of the given table, we can use the formula for the slope (rate of change) between two points on a line:

`\[ \text{Slope (Rate of Change)} = \frac{\text{Change in y (Miles)}}{\text{Change in x (Weeks)}} \]`

From the table provided, we can choose two points to calculate the rate of change. For simplicity, we can use the first and the last points:

- Point 1 (Week 1, Miles 5)

- Point 8 (Week 8, Miles 15.5)

The change in miles (ΔMiles) from week 1 to week 8 is the difference in the Miles values at these two points, and the change in weeks (ΔWeeks) is the difference in the Week values.

Let's calculate the rate of change:

`\[ \text{Rate of Change} = (15.5 - 5)/(8 - 1) \]`

`\[ \text{Rate of Change} = (10.5)/(7) \]`

Now we'll calculate this value to find the rate of change in miles per week.

The rate of change of this function, which represents the number of miles Robert runs per week, is 1.5 miles per week.

Answer 2

Since the relationship is linear, it follows that the rate is given by the slope of the line that passes through any two points in the table.

Let 11, 5 be the first point and (2, 6.5) be the second point.

Therefore,

.