To find the rate of change, we need to determine how much the distance changes for each additional hour of time. We can do this by choosing two points from the table and calculating the difference in distance divided by the difference in time.
Let's take two consecutive points from the table:
- Between the first point (4 hours, 260 miles) and the second point (6 hours, 390 miles):
\[ \text{Change in Distance} = 390 \text{ miles} - 260 \text{ miles} = 130 \text{ miles} \] \[ \text{Change in Time} = 6 \text{ hours} - 4 \text{ hours} = 2 \text{ hours} \] \[ \text{Rate of Change} = \frac{\text{Change in Distance}}{\text{Change in Time}} = \frac{130 \text{ miles}}{2 \text{ hours}} = 65 \text{ miles per hour} \]
- We can verify this rate of change with the next pair of points (6 hours, 390 miles) and (8 hours, 520 miles):
\[ \text{Change in Distance} = 520 \text{ miles} - 390 \text{ miles} = 130 \text{ miles} \] \[ \text{Change in Time} = 8 \text{ hours} - 6 \text{ hours} = 2 \text{ hours} \] \[ \text{Rate of Change} = \frac{130 \text{ miles}}{2 \text{ hours}} = 65 \text{ miles per hour} \]
- We can check one more pair (8 hours, 520 miles) and (10 hours, 650 miles):
\[ \text{Change in Distance} = 650 \text{ miles} - 520 \text{ miles} = 130 \text{ miles} \] \[ \text{Change in Time} = 10 \text{ hours} - 8 \text{ hours} = 2 \text{ hours} \] \[ \text{Rate of Change} = \frac{130 \text{ miles}}{2 \text{ hours}} = 65 \text{ miles per hour} \]
Since we have confirmed the same rate of change across all intervals of time, we can conclude that the rate of change is \( 65 \) miles per hour.
Explanation:
The rate of change of \( 65 \) miles per hour indicates that for every hour that passes, the distance traveled increases by \( 65 \) miles. This consistent rate suggests that the situation describes a scenario where an object or person is moving at a steady speed, such as a vehicle traveling along a road without stopping or changing speed. Thus, over time, the total distance can be predicted simply by multiplying the time by the speed (in this instance, \( 65 \) mph).