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The question provides a scenario where:

- A tractor is 4 miles away from the barn.
- It stays stationary for 1 hour.
- Then, it drives back to the barn at 4 mph.

The task is to find the correct piecewise function \( f(t) \), which describes the tractor's distance from the barn over time \( t \), where \( t \) is in hours.

### Step-by-step breakdown:

1. **From 0 to 1 hour**:
   The tractor remains stationary for the first hour. This means the distance from the barn stays constant at 4 miles. Therefore, for \( 0 \leq t \leq 1 \), \( f(t) = 4 \).

2. **From 1 to 2 hours**:
   After 1 hour, the tractor starts moving back to the barn at a constant speed of 4 mph. The distance decreases as the tractor gets closer to the barn.

   - The tractor starts at a distance of 4 miles.
   - The distance decreases at a rate of 4 miles per hour.
   - The distance \( f(t) \) after time \( t \) can be described by the equation \( f(t) = 4 - 4(t - 1) \) for \( 1 < t \leq 2 \), where \( t - 1 \) represents the time since it started moving.

   Simplifying this gives: 
   \[
   f(t) = 8 - 4t
   \]

### Correct piecewise function:

\[
f(t) = \begin{cases} 
4 & 0 \leq t \leq 1 \\
8 - 4t & 1 < t \leq 2 
\end{cases}
\]

### Comparing with options:

The correct answer matches **Option A**, where the function is:

\[
f(t) = \begin{cases} 
4 & 0 \leq t \leq 1 \\
-4t + 8 & 1 < t \leq 2 
\end{cases}
\]

Would you like further details, or do you have any questions?

### Related questions:
1. How is a piecewise function structured?
2. What is the significance of the constant speed in the second part of the piecewise function?
3. How do you calculate distance when speed and time are given?
4. Why does the distance decrease linearly in the second part of the function?
5. How can we generalize piecewise functions for different scenarios?

**Tip**: When analyzing piecewise functions, always consider how the different conditions affect the variables in each segment of the function.

A tractor is sitting in a field 4 miles from the barn. It sits there for 1 hour before the farmer drives the tractor back to the barn at a constant speed of 4 mph. Which piecewise function best describes the distance from the barn f(t) for t hours?

Solve a real-world problem where a tractor sits 4 miles from a barn, then drives back at a constant speed. The piecewise function f(t) describes the distance over 2 hours. Learn how to model stationary and moving objects with functions. Suitable for students in Grades 8-10.

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