To determine which output has a constant rate of change, we will analyze the rate of change of both the average speed and the distance traveled based on the given data for different days.
Given data:
[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{Day} & \text{Average Speed (mph)} & \text{Distance (mi)} \\
\hline
3 & 55 & 495 \\
\hline
4 & 58 & 660 \\
\hline
5 & 63 & 825 \\
\hline
6 & 65 & 990 \\
\hline
7 & 68 & 1,155 \\
\hline
\end{array}
\][/tex]
### Step 1: Calculating the rate of change for Average Speed
The rate of change of average speed between consecutive days is calculated as follows:
[tex]\[
\begin{align*}
\text{From day 3 to day 4:} & \quad \frac{58 - 55}{4 - 3} = 3 \\
\text{From day 4 to day 5:} & \quad \frac{63 - 58}{5 - 4} = 5 \\
\text{From day 5 to day 6:} & \quad \frac{65 - 63}{6 - 5} = 2 \\
\text{From day 6 to day 7:} & \quad \frac{68 - 65}{7 - 6} = 3 \\
\end{align*}
\][/tex]
The rates of change for average speed are [tex]\([3, 5, 2, 3]\)[/tex], which are not constant.
### Step 2: Calculating the rate of change for Distance
Next, we calculate the rate of change of distance between consecutive days:
[tex]\[
\begin{align*}
\text{From day 3 to day 4:} & \quad \frac{660 - 495}{4 - 3} = 165 \\
\text{From day 4 to day 5:} & \quad \frac{825 - 660}{5 - 4} = 165 \\
\text{From day 5 to day 6:} & \quad \frac{990 - 825}{6 - 5} = 165 \\
\text{From day 6 to day 7:} & \quad \frac{1,155 - 990}{7 - 6} = 165 \\
\end{align*}
\][/tex]
The rates of change for distance are [tex]\([165, 165, 165, 165]\)[/tex], which are constant.
### Conclusion
1. Output with a constant rate of change: The distance traveled has a constant rate of change.
2. Constant rate of change: The constant rate of change is 165 miles per day.
3. Linear relationship: Since the rate of change in distance is constant, the relationship between the days and the distance traveled represents a linear function.
