Investigation strategies

Investigation report

Source of problem
I was helping my friend Ted move by driving his car across country behind the rental truck he was driving. As we started he pointed out that the car needed gas. He said he'd go on ahead because the truck would go only 55 mph and I can catch up with him. After filling the tank I left the gas station 15 minutes behind him. As I drove along at 60 mph, I began to wonder how long it would take to catch up with him.

Problem
How long will a vehicle traveling at 60 mph require to catch up with a vehicle traveling at 55 mph if the slower vehicle has a 15-minute head start?

Analysis
I began by making a guess of 45 minutes. Then I tried to make a table. I kept getting confused about measuring time; then I realized that the problem wasn't clear. Should time be measured starting with the departure of the first vehicle or with the departure of the second? I didn't see that either way was better than the other, but since the question was posed by the driver of the second vehicle I decided to measure time from that point of view. This led to a clarification of the problem.

Problem
A truck starts out driving at 55 mph. Fifteen minutes later a car starts from the same place driving at 60 mph along the same route. How long will the car travel before it catches up with the truck?

Analysis
My table at first stopped at an hour, but as I filled in the miles traveled I realized that the time required would be much more. This led to a solution.

Answer
The time required for the car to catch up with the truck is 165 minutes, or 2.75 hours.
Justification
This table shows how far each vehicle has traveled in increments of 15 minutes after the car starts out.

Minutes after the car starts:	0	15	30	45	60	75	90	105	120	135	150	165
Truck's distance (miles)	13.75	27.5	41.25	55	68.75	82.5	96.25	110	123.75	137.5	151.25	165
Car's distance(miles)	0	15	30	45	69	75	90	105	120	135	150	165

But the solution was messy. It required too much arithmetic. Surely, I thought, the problem could be solved by algebra. I decided to let t represent the unknown, the number of minutes after the car started. Using the variables led to a solution.

Answer:
The time required for the car to catch up with the truck is 165 minutes, or 2.75 hours.
Justification:
Let t represent the number of minutes after the car started. Then the car's distance traveled is t miles. Dimension analysis gives the truck's distance after it started to (t minutes)(55 miles/hour)/(60 minutes/hour), which is about 0.9167t miles. From the table, the distance the truck traveled after the car started was 13.75 + 0.9167t miles. To find the time at which the car caught up with the truck, I set the two expressions equal and solved for t.
t = 13.75 + 0.9167t
0.0833t = 13.75
t = 165.067, which is approximately 165 minutes.

At first I was puzzled by the discrepancy in answers. I realized that the second was due to round-off error, coming from taking the truck's distance to be 0.9167t. The rate is actually 0.916666. . . . Indeed, if I use 0.916667 instead of 0.9167, I get t = 165.00066 minutes.

Critique
I'm pretty happy with the two solutions. I wonder if there's a way to solve the problem algebraically and get the exact answer, though. The two linear expressions remind me of graphs. Could I graph these expressions and get a solution that way?

Analysis
As I went to write up this report and was rewriting the table carefully, I kept seeing the distances for the car and truck getting closer together. That led me to envision the gap between them closing. I wondered if I could express the size of that gap in terms of the time t and then see when the size of the gap reached 0. The gap began at 13.75 miles and then closed. I messed around to come up with an expression in terms of t, and then I realized that the rate at which the gap was closing was 5 miles per hour, the difference between 55 and 60 miles per hour. In fact, the problem is equivalent to this one.

Problem
If a car and a truck are 13.75 miles apart and the truck is sitting still and the car is moving at 5 mph, how long will it take the car to reach the truck?

This problem is easy to solve.

Answer:
2.75 hours, or 165 minutes
Justification:
13.75 miles / (5 miles/hour) = 2.75 hours.

Critique
I like this solution best of all, because it seems more elegant and perhaps even clever. I don't see a need to use algebra at all; simple arithmetic is enough if you look at the problem right. My interest in graphing the expressions also has dwindled. I'm interested, though, in whether or not there are other problems that seem to be algebraic but can be solved by simple arithmetic if looked at in a clever way.