Approximately how high is that peak? - Show Work Please

Question

You are driving along a flat road approaching some mountains. As you pass a sign that informs you that the mountain is 4 miles away, the angle of elevation to the highest peak you can see is 32 degrees. Approximately how high is that peak?

Answer 1

If this is a trigonometry problem:

tan(32°) = x/4

x = 4 tan(32°)

x=2.50 miles

x = 2.5*5280 = 13,197 ft

 

If it's a geometry problem:

The elevation angle is approximately 30 degrees.

The elevation is half the hypotenuse. If the hypotenuse is 2, the vertical leg is 1.

By the Pythagorean theorem, the distance (horizontal leg) is 

√(2²-1²)=√(3)

The ratio of elevation to distance is 1 / √(3)

x/4 = 1 / √(3)

x = (4)/√(3) = 2.31 mi

x = 2.31*5280 = 12,193 ft