To determine the average rate of travel, [tex]\( R \)[/tex], we need to calculate how long it took Mara's family to travel 239.4 miles, and then divide the distance by this travel time. Let's go step-by-step.
1. Convert the start and end times to a single unit, such as minutes:
- The start time is [tex]\( 9:10 \)[/tex] AM. First, convert 9 hours to minutes: [tex]\( 9 \times 60 = 540 \)[/tex] minutes. Then add the 10 minutes. So, the start time in minutes is [tex]\( 540 + 10 = 550 \)[/tex] minutes.
- The end time is [tex]\( 1:40 \)[/tex] PM. Since 1 PM is equivalent to 13 hours in a 24-hour clock, convert 13 hours to minutes: [tex]\( 13 \times 60 = 780 \)[/tex] minutes. Then add the 40 minutes. So, the end time in minutes is [tex]\( 780 + 40 = 820 \)[/tex] minutes.
2. Calculate the total travel time in minutes:
- The total travel time is the difference between the end time and the start time: [tex]\( 820 - 550 = 270 \)[/tex] minutes.
3. Convert the travel time from minutes to hours:
- Since there are 60 minutes in an hour, divide the total travel time by 60 to convert minutes to hours: [tex]\( \frac{270}{60} = 4.5 \)[/tex] hours.
4. Calculate the average rate of travel, [tex]\( R \)[/tex]:
- The average rate of travel, [tex]\( R \)[/tex], is the total distance divided by the total travel time in hours:
[tex]\[
R = \frac{239.4 \text{ miles}}{4.5 \text{ hours}} \approx 53.2 \text{ miles per hour}.
\][/tex]
5. Round the average rate to the nearest whole number:
- So, [tex]\( 53.2 \approx 53 \)[/tex] miles per hour.
Thus, the equation Mara can use to determine their average rate of travel, [tex]\( R \)[/tex], rounded to the nearest mile per hour, is:
[tex]\[
R = \left\lfloor \frac{239.4 \text{ miles}}{4.5 \text{ hours}} + 0.5 \right\rfloor
\][/tex]
Where [tex]\(\left\lfloor x \right\rfloor\)[/tex] denotes rounding the value [tex]\( x \)[/tex] to the nearest whole number. Therefore, their average rate of travel is approximately 53 miles per hour.
