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Difficulty: HardAlgebraic Word Problems and Equation Modeling

An automated freight transport vessel completes a round trip along a 120120-mile coastal canal route from Port Alpha to Port Beta and back. On the return trip from Port Beta to Port Alpha, due to opposing currents, the vessel's constant speed was 2020 miles per hour slower than its constant speed on the trip from Port Alpha to Port Beta. If the total travel time for the entire round trip was 55 hours, what was the average speed of the vessel, in miles per hour, for the entire round trip?

  1. A
    40
  2. 48Answer
  3. C
    50
  4. D
    60
  5. E
    30

Answer

The average speed of the vessel for the entire round trip is 48 miles per hour.
The total distance traveled for the round trip is 120+120=240120 + 120 = 240 miles. Setting up the time equation 120v+120v−20=5\frac{120}{v} + \frac{120}{v-20} = 5 gives the quadratic equation v2−68v+480=0v^2 - 68v + 480 = 0, which factors into (v−60)(v−8)=0(v-60)(v-8) = 0. Since speed must be greater than 2020 mph for a valid return trip, the outgoing speed is 6060 mph and the return speed is 4040 mph. Dividing the total distance of 240240 miles by the total time of 55 hours yields an overall average speed of 4848 miles per hour.

Step-by-Step Solution

1
Define variables and establish time equations for each leg of the journey.
Let vv be the vessel's speed from Port Alpha to Port Beta in miles per hour. The speed from Port Beta to Port Alpha is v−20v - 20 mph. Time outgoing is 120v\frac{120}{v} hours and time returning is 120v−20\frac{120}{v - 20} hours.
Travel time equals distance divided by rate.
2
Set up and simplify the quadratic equation for total travel time.
120v+120v−20=5  ⟹  24v+24v−20=1  ⟹  24(v−20)+24v=v(v−20)  ⟹  v2−68v+480=0\frac{120}{v} + \frac{120}{v - 20} = 5 \implies \frac{24}{v} + \frac{24}{v - 20} = 1 \implies 24(v - 20) + 24v = v(v - 20) \implies v^2 - 68v + 480 = 0.
The sum of the travel times for both legs must equal the total given trip duration of 5 hours.
3
Solve the quadratic equation to determine the valid speed vv.
Factoring (v−60)(v−8)=0(v - 60)(v - 8) = 0 gives roots v=60v = 60 or v=8v = 8. Since the return speed v−20v - 20 must be positive, v=60v = 60 mph.
A speed of v=8v = 8 mph would produce a negative return speed, which is physically invalid.
4
Calculate the average speed for the entire round trip.
Total distance =120+120=240= 120 + 120 = 240 miles. Total time =5= 5 hours. Average speed =2405=48= \frac{240}{5} = 48 miles per hour.
Average speed over a multi-leg trip is defined as total distance divided by total time.

Key Concept

Equation Modeling and Average Speed over Multi-Leg Journeys
Estimated Time:2m 0s
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Topics

Algebraic Word Problems and Equation Modeling

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