To find Lucy's average rate during her trip, we need to determine her speed in miles per hour. This can be calculated using the formula for speed:

$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

Step 1: Calculate the distance traveled

Lucy was 34 miles away after 8 hours and 46 miles away after 11 hours. The difference in distance is:

$46 - 34 = 12 \text{ miles}$

Step 2: Calculate the time taken

The difference in time between the two points is:

$11 - 8 = 3 \text{ hours}$

Step 3: Calculate the average speed

Now, we can use the formula to calculate her speed:

$\text{Speed} = \frac{12 \text{ miles}}{3 \text{ hours}} = 4 \text{ miles per hour}$

Therefore, Lucy's average rate during her trip is 4 miles per hour.

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

Here are 5 related questions for further practice:

1. If Lucy had continued riding for 5 more hours at the same rate, how far would she be from her house?
2. How long would it take Lucy to be 60 miles away from her house at this rate?
3. If Lucy increased her speed by 2 miles per hour, how far would she be in 11 hours?
4. What is the total distance Lucy has covered during the 3-hour time span?
5. If Lucy started 10 miles from her house, how far would she be after 8 hours at the same rate?

Tip: To calculate speed from multiple time and distance measurements, always find the difference between distances and divide by the difference in time.