We are given a table that shows the distance traveled by a car and the car's average speed on different days. We need to analyze whether the rate of change is constant for speed and distance relative to time (days), and identify any linear relationships.
First, let's examine the average speed:
\begin{tabular}{|c|c|c|}
\hline
Day & Average Speed (mph) & Distance (mi) \\
\hline
3 & 55 & 495 \\
\hline
4 & 58 & 660 \\
\hline
5 & 63 & 825 \\
\hline
6 & 65 & 990 \\
\hline
7 & 68 & 1155 \\
\hline
\end{tabular}
### Checking the Rate of Change for Speed:
We calculate the rate of change of speed between consecutive days:
1. From day 3 to day 4: [tex]\((58 - 55) / (4 - 3) = 3\)[/tex] mph per day.
2. From day 4 to day 5: [tex]\((63 - 58) / (5 - 4) = 5\)[/tex] mph per day.
3. From day 5 to day 6: [tex]\((65 - 63) / (6 - 5) = 2\)[/tex] mph per day.
4. From day 6 to day 7: [tex]\((68 - 65) / (7 - 6) = 3\)[/tex] mph per day.
The rates of change in speed are not the same (3, 5, 2, 3 mph). Therefore, the rate of change of speed is not constant.
### Checking the Rate of Change for Distance:
We calculate the rate of change of distance between consecutive days:
1. From day 3 to day 4: [tex]\((660 - 495) / (4 - 3) = 165\)[/tex] miles per day.
2. From day 4 to day 5: [tex]\((825 - 660) / (5 - 4) = 165\)[/tex] miles per day.
3. From day 5 to day 6: [tex]\((990 - 825) / (6 - 5) = 165\)[/tex] miles per day.
4. From day 6 to day 7: [tex]\((1155 - 990) / (7 - 6) = 165\)[/tex] miles per day.
The rates of change in distance are the same (165 miles per day). Therefore, the rate of change of distance is constant.
### Identifying Linear Functions:
A relationship is linear if it has a constant rate of change. Based on our calculations:
- The relationship between time (days) and speed is not linear because the rate of change of speed is not constant.
- The relationship between time (days) and distance is linear because the rate of change of distance is constant.
### Answer Summary:
1. Which output has a constant rate of change?
- The distance (mi) output.
2. What is the constant rate of change?
- The constant rate of change for distance is 165 miles per day.
3. Which relationship represents a linear function?
- The relationship between time (days) and distance (mi) represents a linear function.
