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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 45o, after driving 1 km away from the mountain, the angle between the horizontal and the top of the mountain is 30o.


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


Solution

Consider the diagram:


In the 45o triangle, it is evident that x = h.

In the 30o triangle, tan30o = 1/radical3 = h/(x + 1), leading to x + 1 = radical3h.

Therefore, h + 1 = radical3h, 1 = radical3h minus h, so h(radical3 minus 1) = 1.

Hence, h = 1/(radical3 minus 1) approximately 1.36 km.

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

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