Ashton drove a car, flew in an airplane, and was a passenger on a train to travel from his home in Missouri to his destination in New York. The total distance of the trip was 1,022 miles.
He traveled 874 miles in the airplane at an average speed that was 500 mph faster than the average speed of the car he drove. The train ride was 24 miles and traveled at an average speed 10 mph faster than he drove.
Let r represent the average speed of the car and let h represent the amount of time he spent traveling.
distance rate 1. Use the formula time = to write an equation for the total time of his trip. Express the time as a single fraction.
Onal Practice for
- If he drove at an average rate of 62 mph, how long did the trip take?
- On his return trip, he took a smaller airplane and his average speed was only 530 mph. Assuming that all of the differences in rates stayed the same, how long was his return trip?
- Ashton is reimbursed from his company $25 dollars per hour for his travel time. How much will he receive after the round trip?
The Correct Answer and Explanation is :
Let’s break down the problem step by step.
1. Writing an equation for the total time of the trip:
The total time of the trip is the sum of the times spent in the car, airplane, and train.
We are given that:
- Car: The distance traveled by car is ( 1022 – 874 – 24 = 124 ) miles. Let ( r ) represent the average speed of the car in miles per hour. The time spent in the car is:
[
\text{Time in car} = \frac{124}{r}
] - Airplane: The distance traveled by airplane is 874 miles, and the airplane’s speed is ( r + 500 ) mph. The time spent in the airplane is:
[
\text{Time in airplane} = \frac{874}{r + 500}
] - Train: The distance traveled by train is 24 miles, and the train’s speed is ( r + 10 ) mph. The time spent on the train is:
[
\text{Time in train} = \frac{24}{r + 10}
]
The total time of the trip is the sum of these three times:
[
\text{Total time} = \frac{124}{r} + \frac{874}{r + 500} + \frac{24}{r + 10}
]
This represents the total time spent traveling.
2. If Ashton drove at an average rate of 62 mph, how long did the trip take?
Substitute ( r = 62 ) into the total time equation:
[
\text{Time in car} = \frac{124}{62} = 2 \text{ hours}
]
[
\text{Time in airplane} = \frac{874}{62 + 500} = \frac{874}{562} \approx 1.55 \text{ hours}
]
[
\text{Time in train} = \frac{24}{62 + 10} = \frac{24}{72} = 0.33 \text{ hours}
]
Now add these times together:
[
\text{Total time} = 2 + 1.55 + 0.33 = 3.88 \text{ hours}
]
So the total time for the trip was approximately 3.88 hours.
3. On his return trip, Ashton took a smaller airplane with an average speed of 530 mph. How long was his return trip?
On the return trip, the airplane’s speed is 530 mph, so the time spent in the airplane is:
[
\text{Time in airplane (return)} = \frac{874}{530} \approx 1.65 \text{ hours}
]
The time in the car and train remains the same, so the total time for the return trip is:
[
\text{Time in car} = 2 \text{ hours}
]
[
\text{Time in train} = 0.33 \text{ hours}
]
Now add these times:
[
\text{Total time (return)} = 2 + 1.65 + 0.33 = 3.98 \text{ hours}
]
So the return trip took approximately 3.98 hours.
4. Ashton is reimbursed $25 per hour for his travel time. How much will he receive after the round trip?
The total time for the round trip is the sum of the times for both trips:
[
\text{Total time} = 3.88 + 3.98 = 7.86 \text{ hours}
]
At $25 per hour, the total reimbursement is:
[
\text{Reimbursement} = 7.86 \times 25 = 196.50
]
Ashton will receive $196.50 for his travel time.
Summary:
- The total time for Ashton’s trip was 3.88 hours.
- His return trip took 3.98 hours.
- For the round trip, Ashton will be reimbursed $196.50 for his 7.86 hours of travel time.