Question
Time (hours). 0, 4, 7, 9, 12.
Distance Miles. 0, 248, 449, 565, 757.
The Diaz family went on a road trip for vacation. The table includes a cumulative chart of the time they traveled compared to the distance they drove. Over which interval was their average speed the fastest [0,4], [4,7], [7,9] or [9, 12]? Explain your reasoning by showing steps for finding the average rate of change for each interval.
Over the first interval [0, 4] the average rate of change is _ miles per hour.
Over the second interval [4, 7] the average rate of change is _ miles per hour.
Over the third interval [7, 9] the average rate of change is _ miles per hour.
Over the fourth interval [9, 12] the average rate of change is _ miles per hour.
The family overage the fastest speed of _ miles per hour during the _ interval of the trip.
Distance Miles. 0, 248, 449, 565, 757.
The Diaz family went on a road trip for vacation. The table includes a cumulative chart of the time they traveled compared to the distance they drove. Over which interval was their average speed the fastest [0,4], [4,7], [7,9] or [9, 12]? Explain your reasoning by showing steps for finding the average rate of change for each interval.
Over the first interval [0, 4] the average rate of change is _ miles per hour.
Over the second interval [4, 7] the average rate of change is _ miles per hour.
Over the third interval [7, 9] the average rate of change is _ miles per hour.
Over the fourth interval [9, 12] the average rate of change is _ miles per hour.
The family overage the fastest speed of _ miles per hour during the _ interval of the trip.
Answers
GPT-4o mini
To find the average speed (rate of change) for each interval, we will use the formula for average rate of change (speed) over an interval \([a, b]\):
\[
\text{Average Speed} = \frac{\text{Change in Distance}}{\text{Change in Time}} = \frac{D(b) - D(a)}{T(b) - T(a)}
\]
where \(D\) is the distance traveled and \(T\) is the time.
### Interval [0, 4]
- **Distance at time 0:** 0 miles
- **Distance at time 4:** 248 miles
Calculating the average speed:
\[
\text{Average Speed} = \frac{248 - 0}{4 - 0} = \frac{248}{4} = 62 \text{ miles per hour}
\]
### Interval [4, 7]
- **Distance at time 4:** 248 miles
- **Distance at time 7:** 449 miles
Calculating the average speed:
\[
\text{Average Speed} = \frac{449 - 248}{7 - 4} = \frac{201}{3} = 67 \text{ miles per hour}
\]
### Interval [7, 9]
- **Distance at time 7:** 449 miles
- **Distance at time 9:** 565 miles
Calculating the average speed:
\[
\text{Average Speed} = \frac{565 - 449}{9 - 7} = \frac{116}{2} = 58 \text{ miles per hour}
\]
### Interval [9, 12]
- **Distance at time 9:** 565 miles
- **Distance at time 12:** 757 miles
Calculating the average speed:
\[
\text{Average Speed} = \frac{757 - 565}{12 - 9} = \frac{192}{3} = 64 \text{ miles per hour}
\]
### Summary of Average Speeds
- **[0, 4]:** 62 miles per hour
- **[4, 7]:** 67 miles per hour
- **[7, 9]:** 58 miles per hour
- **[9, 12]:** 64 miles per hour
### Conclusion
The family averaged the fastest speed of **67 miles per hour** during the **[4, 7] interval** of the trip.
\[
\text{Average Speed} = \frac{\text{Change in Distance}}{\text{Change in Time}} = \frac{D(b) - D(a)}{T(b) - T(a)}
\]
where \(D\) is the distance traveled and \(T\) is the time.
### Interval [0, 4]
- **Distance at time 0:** 0 miles
- **Distance at time 4:** 248 miles
Calculating the average speed:
\[
\text{Average Speed} = \frac{248 - 0}{4 - 0} = \frac{248}{4} = 62 \text{ miles per hour}
\]
### Interval [4, 7]
- **Distance at time 4:** 248 miles
- **Distance at time 7:** 449 miles
Calculating the average speed:
\[
\text{Average Speed} = \frac{449 - 248}{7 - 4} = \frac{201}{3} = 67 \text{ miles per hour}
\]
### Interval [7, 9]
- **Distance at time 7:** 449 miles
- **Distance at time 9:** 565 miles
Calculating the average speed:
\[
\text{Average Speed} = \frac{565 - 449}{9 - 7} = \frac{116}{2} = 58 \text{ miles per hour}
\]
### Interval [9, 12]
- **Distance at time 9:** 565 miles
- **Distance at time 12:** 757 miles
Calculating the average speed:
\[
\text{Average Speed} = \frac{757 - 565}{12 - 9} = \frac{192}{3} = 64 \text{ miles per hour}
\]
### Summary of Average Speeds
- **[0, 4]:** 62 miles per hour
- **[4, 7]:** 67 miles per hour
- **[7, 9]:** 58 miles per hour
- **[9, 12]:** 64 miles per hour
### Conclusion
The family averaged the fastest speed of **67 miles per hour** during the **[4, 7] interval** of the trip.