This table shows the distance traveled by a car and the car's average speed on different days.

\begin{tabular}{|c|c|c|}
\hline
Day & \begin{tabular}{c}
Average \\
Speed \\
(mph)
\end{tabular} & \begin{tabular}{c}
Distance \\
(mi)
\end{tabular} \\
\hline
3 & 55 & 495 \\
\hline
4 & 58 & 660 \\
\hline
5 & 63 & 825 \\
\hline
6 & 65 & 990 \\
\hline
7 & 68 & 1,155 \\
\hline
\end{tabular}

Given that the days are the input:

1. Which output has a constant rate of change?
[tex]$\square$[/tex]
2. What is the constant rate of change?
[tex]$\square$[/tex]
3. Which relationship represents a linear function?
[tex]$\square$[/tex]



Answer :

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.