Let's analyze the given data to determine which dependent variable has a constant rate of change and what that constant rate is.
The table gives us the following values for days, average speeds, and distances:
- Days: 3, 4, 5, 6, 7
- Average Speeds (in mph): 55, 58, 63, 65, 68
- Distances (in miles): 495, 660, 825, 990, 1155
First, we need to calculate the rate of change for the average speed and the distance over time. The rate of change between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the formula: [tex]\((y_2 - y_1) / (x_2 - x_1)\)[/tex].
### Rate of Change for Average Speed:
- From Day 3 to Day 4: [tex]\((58 - 55) / (4 - 3) = 3.0\)[/tex] mph/day
- From Day 4 to Day 5: [tex]\((63 - 58) / (5 - 4) = 5.0\)[/tex] mph/day
- From Day 5 to Day 6: [tex]\((65 - 63) / (6 - 5) = 2.0\)[/tex] mph/day
- From Day 6 to Day 7: [tex]\((68 - 65) / (7 - 6) = 3.0\)[/tex] mph/day
The rates of change for average speed are: 3.0, 5.0, 2.0, 3.0 mph/day. Since these values are not constant, the average speed does not have a constant rate of change.
### Rate of Change for Distance:
- From Day 3 to Day 4: [tex]\((660 - 495) / (4 - 3) = 165.0\)[/tex] miles/day
- From Day 4 to Day 5: [tex]\((825 - 660) / (5 - 4) = 165.0\)[/tex] miles/day
- From Day 5 to Day 6: [tex]\((990 - 825) / (6 - 5) = 165.0\)[/tex] miles/day
- From Day 6 to Day 7: [tex]\((1155 - 990) / (7 - 6) = 165.0\)[/tex] miles/day
The rates of change for distance are: 165.0, 165.0, 165.0, 165.0 miles/day. Since these values are constant, the distance traveled has a constant rate of change.
### Conclusion:
The dependent variable that has a constant rate of change is the distance.
The constant rate of change for the distance is 165.0 miles/day.
