https://file.aiquickdraw.com/mathbot/file/1727989217632exq8m1er.jpg

https://file.aiquickdraw.com/mathbot/file/1727989217632ifr8qu1a.jpg

Let's analyze the problem step by step based on the graph provided in the first image.

1. **Part (a) - What is the distance traveled at 0 hours?**

   From the graph, at time 0 hours, the line starts at the origin (0, 0). Therefore, the distance traveled at 0 hours is:
   \[
   0 \text{ miles}.
   \]

2. **Part (b) - What is the distance traveled at 1 hour?**

   By examining the graph at \( t = 1 \) hour, the distance corresponds to approximately 40 miles. Therefore, the distance traveled at 1 hour is:
   \[
   40 \text{ miles}.
   \]

3. **Part (c) - How much does the distance increase for each hour since Dan crossed the bridge?**

   The graph shows a constant increase in distance, which indicates a constant speed. The slope of the line represents the rate of change in distance over time. From the graph, after 2 hours, the distance is 80 miles, and after 3 hours, it is 120 miles. This constant increase shows that the distance increases by:
   \[
   40 \text{ miles per hour}.
   \]

4. **Part (d) - Are the amounts given in parts (b) and (c) equal? Why or why not?**

   Yes, the amounts given in parts (b) and (c) are equal because the line passes through the origin (0, 0). This indicates that Dan was driving at a constant speed from the moment he crossed the bridge, and the distance at 1 hour (40 miles) matches the rate of increase in distance (40 miles per hour). Therefore, the correct answer is:
   \[
   \text{Yes, because the line passes through } (0, 0).
   \]

Would you like more details on any part of the solution? Here are five related questions to deepen understanding:

1. How do you calculate the slope of a line using two points from a graph?
2. What does it mean for a line to pass through the origin in the context of speed and distance?
3. How would the graph change if Dan had a different constant speed?
4. If the line had passed through a different point instead of the origin, what would that indicate about the initial conditions?
5. How can you determine the total distance traveled after 5 hours?

**Tip**: The slope of a distance-time graph represents speed. A steeper slope means a higher speed.

What is the distance traveled at different times, and how does the distance change over time based on the graph provided?

Learn how to interpret a distance-time graph to find the distance traveled at different times and the constant speed. This problem involves calculating the slope of the graph, which represents speed, and understanding linear relationships. Suitable for students in Grades 6-8.

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