Al and Ben are drivers for SD Trucking Company. One snowy day, Ben left SD at 8:00 a.m. heading east...

1. Translate the problem requirements: Ben starts at \(\mathrm{8:00\,a.m.}\) going east at \(\mathrm{20\,mph}\), Al starts at \(\mathrm{11:00\,a.m.}\) going west at \(\mathrm{40\,mph}\). We need to find when their combined distance traveled equals \(\mathrm{240\,miles}\).
2. Set up time relationships: Express how long each person has been driving in terms of the unknown retrieval time.
3. Apply distance relationships: Use the fact that \(\mathrm{distance = speed \times time}\) to express each person's distance traveled.
4. Solve using the total distance constraint: Set up an equation where Al's distance plus Ben's distance equals \(\mathrm{240\,miles}\) and solve for the retrieval time.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening in plain English first. Ben starts his journey at \(\mathrm{8:00\,a.m.}\) driving east at a steady \(\mathrm{20\,miles\,per\,hour}\). Al starts later at \(\mathrm{11:00\,a.m.}\) driving west at \(\mathrm{40\,miles\,per\,hour}\). At some unknown time during the day, the dispatcher checks the system and finds they've driven a combined total of \(\mathrm{240\,miles}\).

The key insight is that Ben gets a 3-hour head start (from \(\mathrm{8:00\,a.m.}\) to \(\mathrm{11:00\,a.m.}\)) before Al even begins driving. After \(\mathrm{11:00\,a.m.}\), both are driving simultaneously until the dispatcher retrieves the data.

Process Skill: TRANSLATE - Converting the story into clear time relationships

2. Set up time relationships

Let's say the dispatcher retrieves the data at time \(\mathrm{T}\) (measured in hours after \(\mathrm{8:00\,a.m.}\)).

Since Ben started at \(\mathrm{8:00\,a.m.}\), Ben has been driving for \(\mathrm{T}\) hours total.

Since Al started at \(\mathrm{11:00\,a.m.}\) (which is 3 hours after \(\mathrm{8:00\,a.m.}\)), Al has been driving for \(\mathrm{(T - 3)}\) hours, but only if \(\mathrm{T}\) is greater than 3. If the dispatcher checked before \(\mathrm{11:00\,a.m.}\), Al wouldn't have driven at all, but since we know both have driven some distance, \(\mathrm{T}\) must be greater than 3.

3. Apply distance relationships

Now we can calculate how far each person has traveled using the simple relationship: \(\mathrm{distance = speed \times time}\).

Ben's distance = \(\mathrm{20\,mph \times T\,hours = 20T\,miles}\)

Al's distance = \(\mathrm{40\,mph \times (T - 3)\,hours = 40(T - 3)\,miles}\)

This makes intuitive sense: Ben has been going slower but for longer, while Al has been going faster but started later.

4. Solve using the total distance constraint

We know their combined distance is \(\mathrm{240\,miles}\), so:

\(\mathrm{Ben's\,distance + Al's\,distance = 240}\)
\(\mathrm{20T + 40(T - 3) = 240}\)

Let's expand this step by step:
\(\mathrm{20T + 40T - 120 = 240}\)
\(\mathrm{60T - 120 = 240}\)
\(\mathrm{60T = 360}\)
\(\mathrm{T = 6}\)

So \(\mathrm{T = 6}\) hours after \(\mathrm{8:00\,a.m.}\), which means the dispatcher retrieved the data at \(\mathrm{2:00\,p.m.}\)

Process Skill: APPLY CONSTRAINTS - Using the total distance requirement to find the exact time

5. Final Answer

Let's verify: At \(\mathrm{2:00\,p.m.}\) (\(\mathrm{6}\) hours after \(\mathrm{8:00\,a.m.}\)):
• Ben has driven for \(\mathrm{6\,hours\,at\,20\,mph = 120\,miles}\)
• Al has driven for \(\mathrm{3\,hours\,at\,40\,mph = 120\,miles}\)
• Total = \(\mathrm{240\,miles}\) ✓

The answer is B. 2:00 p.m.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the time reference point: Students often struggle with setting up the correct time variable. They might use different reference points for Ben and Al (like measuring Ben's time from \(\mathrm{8:00\,a.m.}\) but Al's time from \(\mathrm{11:00\,a.m.}\)) instead of using a single consistent reference point. This leads to incorrect time relationships and makes the problem unsolvable.

2. Missing the head start concept: Students frequently overlook that Ben gets a 3-hour head start before Al begins driving. They might set up the equation assuming both drivers started simultaneously, which would give Ben's time = Al's time, leading to an incorrect equation setup.

3. Confusing the direction constraint: Some students might think that because Ben and Al are driving in opposite directions (east vs west), they need to subtract their distances rather than add them. However, the problem asks for combined total distance traveled, not their separation distance.

Errors while executing the approach

1. Algebraic expansion errors: When expanding \(\mathrm{40(T - 3)}\), students commonly make arithmetic mistakes, such as getting \(\mathrm{40T - 40}\) instead of \(\mathrm{40T - 120}\), or forgetting to distribute the 40 to both terms inside the parentheses.

2. Sign errors in equation solving: Students might make errors when moving terms across the equation. For example, when moving \(\mathrm{-120}\) to the right side, they might forget to change the sign, writing \(\mathrm{60T = 240 - 120 = 120}\) instead of \(\mathrm{60T = 240 + 120 = 360}\).

Errors while selecting the answer

1. Incorrect time conversion: Students might correctly solve for \(\mathrm{T = 6}\) but then misinterpret what this means. Since \(\mathrm{T}\) represents hours after \(\mathrm{8:00\,a.m.}\), the answer is \(\mathrm{8:00\,a.m. + 6\,hours = 2:00\,p.m.}\) However, students might confuse this and think \(\mathrm{T}\) represents hours after \(\mathrm{11:00\,a.m.}\) or some other reference point, leading them to select a different time.