Let's break down the problem into parts:

8(a) - Draw a distance-time graph for her journey.

1. First part of the journey:

   • For the first 30 minutes, Lily is traveling at 40 miles per hour.
   • In 30 minutes (0.5 hours), the distance traveled is: $\text{Distance} = \text{Speed} \times \text{Time} = 40 \times 0.5 = 20 \text{ miles}.$
   • So at 30 minutes, the distance traveled is 20 miles.

2. Stopping for 15 minutes:

   • For the next 15 minutes, Lily is stationary, so the distance doesn't change.
   • The graph should have a flat line from 30 minutes to 45 minutes.

3. Second part of the journey:

   • After the stop, Lily resumes the journey at 60 miles per hour for the last 15 minutes (45 minutes to 60 minutes).
   • In 15 minutes (0.25 hours), the distance traveled is: $\text{Distance} = 60 \times 0.25 = 15 \text{ miles}.$
   • So by 60 minutes, the total distance traveled is $20 + 15 = 35$ miles.

The graph will have:

• A straight line from (0, 0) to (30, 20).
• A flat line from (30, 20) to (45, 20).
• A straight line from (45, 20) to (60, 35).

8(b) - Write down the average speed for the total journey.

The total distance traveled is 35 miles, and the total journey time is 1 hour. The average speed is given by: $\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{35}{1} = 35 \text{ mph}.$

Do you want more details or have any questions?

Here are 5 related questions:

1. What is the relationship between speed, distance, and time?
2. How does the slope of a distance-time graph relate to speed?
3. What happens to the graph when the speed is zero?
4. How would the graph change if Lily increased her speed for the first part of the journey?
5. What if Lily traveled at a constant speed for the entire journey?

Tip: The steeper the line on a distance-time graph, the higher the speed.