# 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?

Jul 25, 2017

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.

$\implies$ $O A$ is the height of the mountain, $O B$ is the distance which the skier cover or the distance between the top of mountain and the lodge and at last $A B$ is the distance between mountain and the lodge which is to be calculated.

In our situation, theta=54^o25^'28" ~=54.4244^o $O B = 8750$ feet, AB=?

From $\Delta O A B$ we have,

$C o s \theta = \frac{A B}{O B}$

$\implies A B = \left(O B\right) C o s \theta = \left(8750\right) \cos \left({54.4244}^{o}\right) = \left(8750\right) \left(0.58178\right) = 5090.575$ feet.

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