[ Motion problems ]

The cabins of the ski lift are numbered consecutively from 1 to 99. Igor got into cabin No. 42 at the foot of the mountain and at some point noticed that he was level with cabin No. 13 moving downwards (see figure), and 15 seconds later his cabin was level with cabin No. 12.

How long will it take for Igor to reach the top of the mountain?

![](https://cdn.mathpix.com/cropped/2024_05_06_f57e4fb73adf3cbf0752g-35.jpg?height=591&width=203&top_left_y=-1&top_left_x=919)
Since Igor first caught up with cabin No. 13 and then with cabin No. 12, the numbering goes in the direction of the lift's movement. Let's assume the distance between adjacent cabins is one unit. Then the distance along the cable between cabins No. 42 and No. 12 is 69 units: 57 units to cabin No. 99 and another 12 units to cabin No. 12. Therefore, the distance between Igor and the mountain peak is half of this amount, which is 34.5 units.

Since the cabins that cabin No. 42 catches up with are moving towards it at the same speed, the closing speed of the cabins is twice the speed of the lift. This means it takes 30 seconds for one cabin to cover one unit of distance, and Igor will reach the mountain peak in \(34.5 \cdot 30: 60 = 17.25\) (minutes).

## Answer

In 17 minutes and 15 seconds.