To find Anna's average speed over the first 6 hours, we need to follow these steps:
1. Identify the total distance traveled in the first 6 hours.
2. Determine the total time elapsed during this period.
3. Calculate the average speed by dividing the total distance by the total time.
Step 1: Identify the total distance traveled in the first 6 hours.
From the given table:
- At [tex]\( t = 0 \)[/tex] hours, the distance [tex]\( D(0) \)[/tex] is 10 miles.
- At [tex]\( t = 6 \)[/tex] hours, the distance [tex]\( D(6) \)[/tex] is 292 miles.
The total distance traveled in the first 6 hours is:
[tex]\[
\text{Distance covered} = D(6) - D(0) = 292 \text{ miles} - 10 \text{ miles} = 282 \text{ miles}
\][/tex]
Step 2: Determine the total time elapsed during this period.
The total time is from [tex]\( t = 0 \)[/tex] hours to [tex]\( t = 6 \)[/tex] hours, which is:
[tex]\[
\text{Total time} = 6 \text{ hours}
\][/tex]
Step 3: Calculate the average speed.
The average speed is calculated by dividing the total distance by the total time:
[tex]\[
\text{Average speed} = \frac{\text{Distance covered}}{\text{Total time}} = \frac{282}{6} = 47
\][/tex]
Thus, Anna's average speed over the first 6 hours is:
[tex]\[
\boxed{47}
\][/tex]
