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Ai Mi is going to drive from her house to City A without stopping. Let D represent Ai Mi's distance from City A t hours after leaving her house. A graph of D is shown below. Write an equation for D then state the slope of the graph and determine its interpretation in the context of the problem.

What is the D=

What is the slope of the function

To analyze the graph and answer your question, let's break it down:

1. **Identify the important points on the graph**: 
   - At \( t = 0 \) hours (when Ai Mi starts driving), the distance remaining to City A is 300 miles (point \( (0, 300) \)).
   - At \( t = 5 \) hours, the distance remaining is 0 miles (point \( (5, 0) \)).

2. **Find the slope of the graph**: 
   The slope (\( m \)) is calculated as the change in distance (\( \Delta D \)) over the change in time (\( \Delta t \)).

   \[
   m = \frac{\Delta D}{\Delta t} = \frac{0 - 300}{5 - 0} = \frac{-300}{5} = -60
   \]

   So, the slope is \( -60 \).

3. **Interpretation of the slope**: 
   The slope of \( -60 \) means that Ai Mi is driving at a constant speed of 60 miles per hour towards City A. The negative sign indicates that the distance to City A is decreasing.

4. **Equation for \( D \)**: 
   The general form of a linear equation is \( D = mt + b \), where:
   - \( m \) is the slope (which we've determined to be \( -60 \)),
   - \( b \) is the y-intercept (which is the initial distance from City A, 300 miles when \( t = 0 \)).

   Thus, the equation for \( D \) is:

   \[
   D = -60t + 300
   \]

This equation represents the distance remaining to City A \( t \) hours after Ai Mi starts driving.

### Summary:
- **Equation**: \( D = -60t + 300 \)
- **Slope**: \( -60 \) (indicating Ai Mi is driving 60 miles per hour towards City A)

Would you like further details on any aspect of this, or have any additional questions?

### Follow-up questions:
1. What would be the distance remaining after 2 hours of driving?
2. How long would it take Ai Mi to reach halfway (150 miles) to City A?
3. How far will Ai Mi be from City A after driving for 3.5 hours?
4. If Ai Mi drives slower, say 50 miles per hour, how would the equation and graph change?
5. Can the same method be used to find the equation if the trip were to take 7 hours instead of 5?

### Tip:
When interpreting slopes in real-world contexts, pay attention to units (like miles per hour) and the direction (positive or negative) to understand what they mean practically.

Ai Mi is going to drive from her house to City A without stopping. Let D represent Ai Mi's distance from City A t hours after leaving her house. A graph of D is shown below. Write an equation for D, then state the slope of the graph and determine its interpretation in the context of the problem.

Explore how to write the equation for Ai Mi's distance from City A as a function of time, interpret the slope of the graph, and understand the constant speed at which Ai Mi drives towards her destination. Suitable for Grades 6-8 students.

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