To calculate the total distance traveled between the three locations (sledding, ice skating, and skiing), we need to find the distances between each pair of points using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Given the coordinates:
- Sledding: (2, 12)
- Ice Skating: (2, -4)
- Skiing: (-4, -8)
Step 1: Calculate the distance from Sledding to Ice Skating
Let \( (x_1, y_1) = (2, 12) \) and \( (x_2, y_2) = (2, -4) \).
\[ d_{\text{sledding to ice skating}} = \sqrt{(2 - 2)^2 + (-4 - 12)^2} \] \[ = \sqrt{0 + (-16)^2} \] \[ = \sqrt{256} = 16 \]
Step 2: Calculate the distance from Ice Skating to Skiing
Let \( (x_1, y_1) = (2, -4) \) and \( (x_2, y_2) = (-4, -8) \).
\[ d_{\text{ice skating to skiing}} = \sqrt{(-4 - 2)^2 + (-8 + 4)^2} \] \[ = \sqrt{(-6)^2 + (-4)^2} \] \[ = \sqrt{36 + 16} \] \[ = \sqrt{52} = 2\sqrt{13} \approx 7.21 \]
Step 3: Calculate the total distance traveled
Now, sum the two distances:
\[ \text{Total Distance} = 16 + 2\sqrt{13} \]
Approximate Value
\[ 2\sqrt{13} \approx 7.21 \] So,
\[ \text{Total Distance} \approx 16 + 7.21 \approx 23.21 \]
Conclusion
The total distance traveled is approximately 23.21 miles.