To find the interval where the Diaz family had the fastest average speed, we need to calculate the average speed for each time interval given, using the formula:
\[ \text{Average Speed} = \frac{\text{Change in Distance}}{\text{Change in Time}} \]
Let's calculate the average speed for each interval:
-
Interval [0, 4]:
- Change in distance = 248 miles - 0 miles = 248 miles
- Change in time = 4 hours - 0 hours = 4 hours
- Average speed = \( \frac{248}{4} = 62 \) miles per hour
-
Interval [4, 7]:
- Change in distance = 449 miles - 248 miles = 201 miles
- Change in time = 7 hours - 4 hours = 3 hours
- Average speed = \( \frac{201}{3} = 67 \) miles per hour
-
Interval [7, 9]:
- Change in distance = 565 miles - 449 miles = 116 miles
- Change in time = 9 hours - 7 hours = 2 hours
- Average speed = \( \frac{116}{2} = 58 \) miles per hour
-
Interval [9, 12]:
- Change in distance = 757 miles - 565 miles = 192 miles
- Change in time = 12 hours - 9 hours = 3 hours
- Average speed = \( \frac{192}{3} = 64 \) miles per hour
Now, let's summarize the average speeds for each interval:
- [0, 4]: 62 mph
- [4, 7]: 67 mph
- [7, 9]: 58 mph
- [9, 12]: 64 mph
The fastest average speed occurred in the interval [4, 7] where the average speed was 67 miles per hour.
Therefore, the answer is:
[4, 7]