To find the height of a mountain, surveyors often find the angle of elevation to the top from two points at the same altitude a fixed distance apart. suppose that the angles of the elevation from two points 500 meters apart are 35. degrees and 25.7 degrees. how high is the mountain above the altitude of the two points

756.36 meters

Step-by-step explanation:

Draw a diagram.  Let's call the horizontal distance from the top of the mountain to the closest point x.

Using tangent = opposite / adjacent, we can write two equations:

tan 35.3° = h / x

tan 25.76° = h / (x + 500)

Solve for x in the first equation and substitute into the second.

x = h / tan 35.3°

tan 25.76° = h / ((h / tan 35.3°) + 500)

Solve for h.

tan 25.76° (h / tan 35.3°) + 500 tan 25.76° = h

500 tan 25.76° = h (1 − (tan 25.76° / tan 35.3°))

500 tan 25.76° tan 35.3° = h (tan 35.3° − tan 25.76°)

h = 500 tan 25.76° tan 35.3° / (tan 35.3° − tan 25.76°)

h ≈ 756.36 meters