## Measuring Mountains

#### Problem

A geographical surveyor places a marker at a position such that the angle between the horizontal and the top of a mountain is 45^{o}, after driving 1 km away from the mountain, the angle between the horizontal and the top of the mountain is 30^{o}.

Use the given information to find the height of the mountain.

#### Solution

Consider the diagram:

In the 45^{o} triangle, it is evident that `x` = `h`.

In the 30^{o} triangle, tan30^{o} = 1/3 = `h`/(`x` + 1), leading to `x` + 1 = 3`h`.

Therefore, `h` + 1 = 3`h`, 1 = 3`h` `h`, so `h`(3 1) = 1.

Hence, `h` = 1/(3 1) 1.36 km.

Problem ID: 132 (Nov 2003) Difficulty: 2 Star