Final answer:
Two possible sets of lengths are 10 km, 12 km, 12 km, 14 km, and another is 12 km, 12 km, 12 km, 14 km. The mode is 12 km and the range is 5 km in both cases, with 12 km being the most frequent length.
Step-by-step explanation:
The student's question involves finding two possible sets of three lengths when the fourth length is 14 km, the mode of the lengths is 12 km, and the range is 5 km. The range of 5 km means that the difference between the smallest and the largest length is 5 km. Since one of the lengths is given as 14 km and the mode is 12 km, 12 must occur more than once.
Here are two possible solutions:
- The lengths could be 10 km, 12 km, 12 km, and 14 km.
- Another possibility is 12 km, 12 km, 12 km, and 14 km.
In the first set, we have 10 as the smallest length and 14 as the largest, which gives us the range of 5 km. The mode is 12 km as it appears twice. In the second set, even though we have three 12 km lengths, the range of 5 km is still maintained with 14 km as the largest length and 12 km (the mode) as the smallest length.