George's home is 20 miles from his place of work. One day as he is driving home from work, his...

1. Translate the problem requirements: Clarify George's journey - he's 20 miles from home when at work, breaks down 5 miles from work (so 15 miles from home), and walks toward home at 4 mph. We need to express his distance from home as a function of walking time.
2. Establish George's starting position: Determine exactly where George begins walking in relation to his home.
3. Model the walking progression: Set up how his distance from home changes as he walks toward it over time.
4. Construct the distance function: Combine starting position and walking progress to create the equation \(\mathrm{d = f(t)}\).

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding exactly what's happening in George's journey. Think of this like following someone on a map:

• George's workplace and home are 20 miles apart
• He's driving home from work when his car breaks down
• The breakdown happens when he's only 5 miles away from work
• This means he still has to cover the remaining distance to get home
• He decides to walk the rest of the way at 4 miles per hour

We need to create a formula that tells us how far George is from home at any given time while he's walking. The key insight here is that we're tracking his distance FROM HOME, not from work.

Process Skill: TRANSLATE - Converting the story into spatial relationships we can work with mathematically

2. Establish George's starting position

Now let's figure out exactly where George begins his walk. This is like finding someone's location on a number line:

• Total distance from work to home = 20 miles
• George's car breaks down 5 miles from work
• So the remaining distance from breakdown point to home = \(\mathrm{20 - 5 = 15}\) miles

This means when George starts walking (at time t = 0), he is exactly 15 miles away from home. This will be our starting point in our equation.

3. Model the walking progression

Now let's think about what happens as George walks. Imagine watching him walk toward home:

• He walks at 4 miles per hour toward his home
• After 1 hour of walking, he covers 4 miles
• After 2 hours of walking, he covers 8 miles
• After t hours of walking, he covers \(\mathrm{4t}\) miles

The key insight: as George walks TOWARD home, his distance FROM home gets smaller. Every step he takes reduces the gap between him and home.

Process Skill: VISUALIZE - Seeing how the distance shrinks as George approaches home

4. Construct the distance function

Now we can put together our complete picture:

• George starts 15 miles from home
• After walking for t hours, he has covered \(\mathrm{4t}\) miles toward home
• So his distance from home = his starting distance - the distance he has walked

In plain English: Distance from home = Starting distance - Distance walked

Using our numbers: \(\mathrm{d = 15 - 4t}\)

Let's verify this makes sense:

• When t = 0 (just started walking): \(\mathrm{d = 15 - 4(0) = 15}\) miles from home ✓
• When t = 1 (after 1 hour): \(\mathrm{d = 15 - 4(1) = 11}\) miles from home ✓
• When t = 3.75 (after 3.75 hours): \(\mathrm{d = 15 - 4(3.75) = 0}\) miles from home (he's arrived!) ✓

4. Final Answer

The equation that gives George's distance from home in terms of his walking time is:

\(\mathrm{d = 15 - 4t}\)

This matches answer choice A. The equation captures that George starts 15 miles from home and gets 4 miles closer to home with each hour of walking.

Common Faltering Points

Errors while devising the approach

The question asks for d to represent "his distance from home," but students often set up their equation thinking about distance from work instead. This leads them to incorrectly model George's position relative to his workplace rather than his house, resulting in equations like \(\mathrm{d = 5 + 4t}\) (thinking he starts 5 miles from work and walks away from it).

Students frequently assume George starts walking from work (20 miles from home) rather than carefully reading that his car breaks down when he's "5 miles from work." This leads to setting up the equation as \(\mathrm{d = 20 - 4t}\) instead of correctly identifying that he starts walking from a position 15 miles from home.

Some students recognize George starts 15 miles from home but incorrectly think that as time passes, his distance from home increases. This conceptual error about direction leads them to write \(\mathrm{d = 15 + 4t}\), not realizing that walking toward home should decrease the distance from home.

Errors while executing the approach

Even when students understand the setup correctly, they may make a simple subtraction error when calculating George's starting distance from home: \(\mathrm{20 - 5}\). Some might incorrectly calculate this as 25 or other wrong values, leading to equations with incorrect constants.

Errors while selecting the answer

No likely faltering points - once students correctly derive \(\mathrm{d = 15 - 4t}\) through proper execution, the answer choice A clearly matches their result.