- The original daily distance is \(\frac{14}{3}\) miles per day.
- In the new scenario, the daily distance becomes \(\frac{1}{2} \times \frac{14}{3} = \frac{7}{3}\) miles per day.
- The trip duration becomes \(3 \times 2 = 6\) days.
- The new total distance traveled is \(\frac{7}{3} \times 6 = 14\) miles, and the new daily distance is \(\boxed{\frac{7}{3}}\) miles.
Explanation
- Calculating the original daily distance
The Miller family paddled a total distance of 14 miles over the course of 3 days, with the same distance covered each day. To find the daily distance \(d\), we divide the total distance by the number of days:
\[d = \frac{14 \text{ miles}}{3 \text{ days}} = \frac{14}{3} \text{ miles/day} \approx 4.67 \text{ miles/day}.\]
- Calculating the new daily distance and time duration
In the new scenario, the daily distance is halved, and the total number of days is doubled.
The new daily distance \(d_{new}\) is:
\[d_{new} = \frac{1}{2} \times d = \frac{1}{2} \times \frac{14}{3} = \frac{7}{3} \text{ miles/day} \approx 2.33 \text{ miles/day}.\]
The new duration of the trip \(t_{new}\) is:
\[t_{new} = 2 \times 3 \text{ days} = 6 \text{ days}.\]
- Calculating the new total distance and final daily distance
To find the total distance \(D_{total\_new}\) for the new scenario, we multiply the new daily distance by the new number of days:
\[D_{total\_new} = d_{new} \times t_{new} = \left(\frac{7}{3} \text{ miles/day}\right) \times 6 \text{ days} = 7 \times 2 \text{ miles} = 14 \text{ miles}.\]
The total distance traveled remains the same, which is 14 miles. The new daily distance is \(\frac{7}{3}\) miles.
Examples
This problem illustrates the concept of inverse variation in a practical context. When one variable is halved (daily distance) and another is doubled (time), the product (total distance) remains constant. This principle is widely used in resource management, such as planning work shifts where reducing the speed of individual tasks while increasing the time allowed results in the same total output, ensuring that deadlines are met even if the pace changes.