A truck left City M and traveled toward City N. A second truck left City N exactly 1 hour later...

1. Translate the problem requirements: Two trucks start from different cities and travel toward each other. The first truck (50 mph) gets a 1-hour head start before the second truck (60 mph) begins. When they meet, their combined distance traveled equals 490 miles. We need to find how far the first truck traveled.
2. Account for the head start advantage: Calculate how far the first truck travels during its 1-hour head start before the second truck even begins moving.
3. Set up the meeting point scenario: Determine how long both trucks travel together from when the second truck starts until they meet, using the fact that they cover the remaining distance at their combined speed.
4. Calculate the first truck's total distance: Add the head start distance to the distance traveled during the time both trucks were moving to find the total distance traveled by the first truck.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what's happening in everyday terms. We have two trucks starting from different cities and driving toward each other - like two friends walking toward each other from opposite ends of a long hallway.

Here are the key facts we need to work with:

• First truck starts from City M, travels at 50 mph
• Second truck starts from City N, travels at 60 mph
• The second truck starts exactly 1 hour after the first truck
• When they meet, their combined distance traveled equals 490 miles
• We need to find how far the first truck traveled

The crucial insight is that the first truck gets a "head start" of 1 hour before the second truck even begins moving.

Process Skill: TRANSLATE - Converting the problem scenario into clear mathematical relationships

2. Account for the head start advantage

During that first hour, while the second truck is still parked, the first truck is already moving. Let's calculate how much ground it covers during this head start.

\(\mathrm{Distance} = \mathrm{Speed} \times \mathrm{Time}\)
Head start distance = \(50 \text{ mph} \times 1 \text{ hour} = 50 \text{ miles}\)

So when the second truck finally starts moving, the first truck is already 50 miles away from City M. This means the trucks now need to cover the remaining distance: \(490 - 50 = 440 \text{ miles}\) between them.

3. Set up the meeting point scenario

Now both trucks are moving toward each other. Think of it this way: they're working together to close the remaining 440-mile gap between them.

When two objects move toward each other, we can think of their combined speed as how fast they're closing the distance between them:
Combined speed = \(50 \text{ mph} + 60 \text{ mph} = 110 \text{ mph}\)

To find how long they travel together before meeting:
\(\mathrm{Time} = \mathrm{Distance} \div \mathrm{Speed}\)
Time together = \(440 \text{ miles} \div 110 \text{ mph} = 4 \text{ hours}\)

Process Skill: VISUALIZE - Seeing the relative motion as a combined closing speed

4. Calculate the first truck's total distance

Now we can find the total distance traveled by the first truck by adding two parts:

Part 1: Distance during head start = 50 miles (calculated in step 2)
Part 2: Distance during the 4 hours both trucks were moving = \(50 \text{ mph} \times 4 \text{ hours} = 200 \text{ miles}\)

Total distance traveled by first truck = \(50 + 200 = 250 \text{ miles}\)

Let's verify this makes sense:

• First truck: 250 miles
• Second truck: \(60 \text{ mph} \times 4 \text{ hours} = 240 \text{ miles}\)
• Combined: \(250 + 240 = 490 \text{ miles}\) ✓

Final Answer

The first truck traveled 250 miles when the trucks passed each other.

This matches answer choice D. 250.

Common Faltering Points

Errors while devising the approach

Students often miss that the second truck starts "exactly 1 hour later" than the first truck. They might assume both trucks start simultaneously, which would lead to setting up equations where both trucks travel for the same amount of time. This crucial detail about the staggered start times is essential for correctly modeling the problem.

Some students might interpret "combined total of 490 miles" as the distance between the two cities, rather than understanding it as the sum of distances traveled by both trucks when they meet. This misinterpretation would lead to incorrect equation setup and wrong calculations.

Students might struggle with visualizing that when two objects move toward each other, their speeds effectively add up to determine how quickly they close the gap between them. Instead, they might try to solve using more complex position equations without recognizing the simpler "combined speed" approach.

Errors while executing the approach

Even when students correctly calculate the 50-mile head start and the 200 miles traveled during the 4 hours both trucks were moving, they might forget to add these two distances together. They could mistakenly report just the 200 miles (distance during joint travel) as their final answer.

When calculating \(440 \div 110 = 4 \text{ hours}\), students might make computational mistakes, especially if working under time pressure. Getting this division wrong would cascade into incorrect final calculations for the distance traveled.

Students might correctly find that the trucks travel together for 4 hours but then mistakenly use 5 hours (adding the 1-hour head start) when calculating the first truck's distance during the joint travel phase, leading to \(50 \text{ mph} \times 5 \text{ hours} = 250 \text{ miles}\) for just this portion.

Errors while selecting the answer

After calculating both distances (first truck: 250 miles, second truck: 240 miles), students might accidentally select 240 miles as their answer, confusing which truck's distance the question is asking for.

Students might report intermediate results like 200 miles (the distance the first truck traveled during joint motion only) or 240 miles (the combined speed × time), rather than the complete 250 miles that includes the head start distance.