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George's home is 20 miles from his place of work. One day as he is driving home from work, his car breaks down when he is 5 miles from work. He starts walking toward his home at an average rate of 4 miles per hour. If \(\mathrm{d}\) represents his distance from home, in miles, and \(\mathrm{t}\) represents the time, in hours, that he has been walking, which of the following gives \(\mathrm{d}\) in terms of \(\mathrm{t}\) ?
Let's start by understanding exactly what's happening in George's journey. Think of this like following someone on a map:
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
Now let's figure out exactly where George begins his walk. This is like finding someone's location on a number line:
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.
Now let's think about what happens as George walks. Imagine watching him walk toward home:
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
Now we can put together our complete picture:
In plain English: Distance from home = Starting distance - Distance walked
Using our numbers: \(\mathrm{d = 15 - 4t}\)
Let's verify this makes sense:
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.
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).
2. Misunderstanding George's starting position for walkingStudents 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.
3. Incorrect direction modeling (addition instead of subtraction)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.
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.
No likely faltering points - once students correctly derive \(\mathrm{d = 15 - 4t}\) through proper execution, the answer choice A clearly matches their result.