Two cars travel at constant rates. The rate at which car [tex]$M$[/tex] travels can be modeled by the equation [tex]$y=50x$[/tex], where [tex]$y$[/tex] is the number of miles traveled given the number of hours spent traveling, [tex]$x$[/tex].

The table represents the travel for car [tex]$P$[/tex].
[tex]\[
\begin{tabular}{|c|c|}
\hline
\begin{tabular}{c}
Time Traveled \\
(hours)
\end{tabular} & \begin{tabular}{c}
Distance Traveled \\
(miles)
\end{tabular} \\
\hline
2 & 90 \\
\hline
4 & 180 \\
\hline
\end{tabular}
\][/tex]

Compare the rates at which the two cars travel. Select from the drop-down lists to complete the sentences correctly.

Car [tex]$M$[/tex] travels at a [tex]$\square$[/tex] rate than car [tex]$P$[/tex]. Every hour, car [tex]$M$[/tex] will travel [tex]$\square$[/tex] miles [tex]$\square$[/tex] than car [tex]$P$[/tex].



Answer :

Let's analyze the information given and compare the rates at which cars M and P travel.

### Step 1: Determine the rate of car M
The rate at which car M travels can be modeled by the equation [tex]\( y = 50x \)[/tex], where [tex]\( y \)[/tex] is the distance traveled and [tex]\( x \)[/tex] is the time in hours. Here, the coefficient of [tex]\( x \)[/tex] in the equation 50 indicates that car M travels 50 miles per hour.

### Step 2: Determine the rate of car P
To find the rate (speed) at which car P travels, we need to use the data provided in the table:

[tex]\[ \begin{array}{|c|c|} \hline \text{Time Traveled (hours)} & \text{Distance Traveled (miles)} \\ \hline 2 & 90 \\ \hline 4 & 180 \\ \hline \end{array} \][/tex]

We can calculate the rate of car P for each row in the table and then average these rates.

From the table:
- At 2 hours, car P travels 90 miles. Thus, the rate is:
[tex]\[ \text{Rate}_1 = \frac{90\text{ miles}}{2\text{ hours}} = 45 \text{ miles per hour} \][/tex]

- At 4 hours, car P travels 180 miles. Thus, the rate is:
[tex]\[ \text{Rate}_2 = \frac{180\text{ miles}}{4\text{ hours}} = 45 \text{ miles per hour} \][/tex]

Since both rates are the same, the average rate at which car P travels is:
[tex]\[ \text{Rate}_P = \frac{45 + 45}{2} = 45 \text{ miles per hour} \][/tex]

### Step 3: Compare the rates
Now, we can compare the rates of car M and car P.

- Car M travels at 50 miles per hour.
- Car P travels at 45 miles per hour.

Clearly, car M travels faster than car P. The difference in their rates is:
[tex]\[ 50 \text{ miles per hour} - 45 \text{ miles per hour} = 5 \text{ miles per hour} \][/tex]

### Conclusion
Using the information we calculated, we can complete the sentences as follows:

Car [tex]\( M \)[/tex] travels at a faster rate than car [tex]\( P \)[/tex]. Every hour, car [tex]\( M \)[/tex] will travel 5 miles more than car [tex]\( P \)[/tex].