Exam Tomorrow - A train leaves San Francisco to Los Angeles..?
Hi All,
We had to write our own exam and I'm not sure if the answer given to me for the question below is correct - can you please answer it including the steps it took to get there? Thanks!
A train leaves San Francisco to Los Angeles, 400 miles away at noon averaging 60 mph. Another train leaves Los Angeles to San Francisco on an adjacent set of tracks at 2pm and averages 50 mph. At what time do they meet?? The answer given is 2:23 but I'm not sure how that works. We are using Distance = Rate x Time if it matters.
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Verified answer
LA train:
D1 = R1 x T
T = Time traveled by LA train before it meets SF train
D1 = distance traveled by LA train
R1 = speed = 50 mph
SF train:
D2 = R2 x (T + 2)
T + 2 = time traveled by SF train (since it started 2 hours early)
D2 = distance traveled by SF train
R2 = speed = 60
But D1 + D2 = 400 miles
R1 x T + R2 x (T + 2) = 400
50T + 60 (T + 2) = 400
110T + 120 = 400
T = (400 - 120)/110 = 2.545 hours
or 2 hours + 0.545 x 60 minutes/hour = 2 hours + 32 minutes + 42 seconds
Since the LA train started at 2 PM plus the time traveled T = 2 hours and 32 minutes, the trains meet at 4:32 PM.
Distance = Rate X Time
Time = Distance/Rate
Rate = Distance/Time
The thing these vehicles have in common is Time and Distance, certainly not Rate.
From San Francisco, it takes 400miles/60 miles/hour, or 6 hours and 40 minutes (6.666... hours) to get to LA.
From LA, it takes 400miles/50 miles/hour, or 8 hours to get to San Francisco.
But the LA-bound leaves at 12:00 noon and the SF-bound leaves two hours later, at 2:00 p.m.
Question 1: Where is the LA-bound at 2:00? Answer: 2 hours X 60 miles/hour = 120 miles. 400 miles - 120 miles = 280 miles from LA when the SF-bound leaves.
Question 2: What is the combined speed of the two trains once both are traveling at the same time? Answer: 50mph + 60mph = 110mph.
Question 3: Since two vehicles approaching each other at a combined speed of 110mph is the same as one vehicle moving at 110mph, how much time would it take one vehicle to cover 280 remaining miles if the total speed is 110 mph? Answer: Time = 280 miles/110 miles/hour = 280/110 hours = 2.54... hours.
Question 4: At what time do the vehicles meet? Answer: 2:00p.m. + 2.54 hours = 4:32:44 p.m. ~ 4:33 p.m.
4:33 p.m.
This is one of those tricky problems because it refers to the outside world. This really could happen.
They meet after x hours, which is x hours after 2 pm.
The 60 mph train has been on its way for an extra two hours
so 60*(x+2) + 50*x = 400
60x + 120 + 50x = 400
110x + 120 = 400
110x =400 - 120
110x = 280 miles
x = 280/110mph (relatively speaking).
x = 2.55 about. I don't think I've made an error, but I'll concede it if I have.
That is the amount of time that it takes for them to meet.
The time that they meet is at 4 oclock and 33 minutes.
.55 of an hour is 33 minutes (.55*60). The answer you were given is not quite correct: it should have added it on to 2 pm. That's the question.