Final answer:
Using the total distance traveled and the total time taken, we set up equations for the time of the outbound and return trips, then solved for the average speed. The correct answer is option B: 18 mph for the outbound trip and 14 mph for the return trip.
Step-by-step explanation:
To solve for the bicyclist's average speed on each part of the trip, we will use the formula for average speed, which is the total distance traveled divided by the total time taken. Let's denote the average speed for the outbound trip as v mph. Thus, for the return trip, the average speed would be v - 4 mph.
The total distance for the round trip is 117 miles each way, so the total distance is 2 x 117 = 234 miles. The total time for the round trip is given as 22 hours.
We can now set up two equations based on the information given:
- Time for outbound trip = Distance / Speed = 117/v hours
- Time for return trip = Distance / Speed = 117/(v - 4) hours
The sum of these two times equals the total trip time:
117/v + 117/(v - 4) = 22
To solve this equation, find a common denominator and combine the terms:
(117(v - 4) + 117v) / (v(v - 4)) = 22
Now we multiply both sides by v(v - 4) to get rid of the fraction, which gives us:
117v - 468 + 117v = 22v^2 - 88v
Combine like terms:
22v^2 - 322v + 468 = 0
Using the quadratic formula or factoring, we find that v = 18 mph as the average speed for the outbound trip.
Therefore, the average speed for the return trip is v - 4 = 14 mph. So option B is the correct choice.