Question

A skier is at the top of a mountain. His angle of depression looking down at the ski lodge is 54°25'28". If he rides the ski lift 8750 feet to the lodge, how far is the lodge from the base of the mountain?

Answer 1

The lodge is `5090.575` feet away from the botton of the mountain.

Explanation:

In the figure above, O is the top of the mountain where the skier is; A is the base of mountain, B is the lodge, `theta` which is nearest to O is the angle of depression. Using property of alternate angles we conclude that the angle at B must be same as the angle of depression.

`implies` `OA` is the height of the mountain, `OB` is the distance which the skier cover or the distance between the top of mountain and the lodge and at last `AB` is the distance between mountain and the lodge which is to be calculated.

In our situation, `theta=54^o25^'28" ~=54.4244^o` `OB=8750` feet, `AB=?`

From `Delta OAB` we have,

`Costheta=(AB)/(OB)`

`implies AB=(OB) Costheta=(8750) cos(54.4244^o)=(8750)(0.58178)=5090.575` feet.

Hence, the lodge is `5090.575` feet away from the botton of the mountain.