A survey team is trying to estimate the height of a mountain above a level plain. From one point on the plain, they observe that the angle of elevation to the top of the mountain is 25 degrees. From a point 1225 feet closer to the mountain along the plain, they find that the angle of elevation is 30 degrees.

Question

How high (in feet) is the mountain?

Answer 1

make a sketch
let the top of mountain be T
base of mountain R
so we want TR

Let A be the point farthest from the top, let B be the closer point,
AB = 1225
In triangle TAB
angle A = 25°
angle TBR = 30° , so angle TBA =150°
leaving us with angle ATB = 5°
by the sine law:
BT/sin25 = 1225/sin 5°
BT = 1225sin25/sin5 = 5940.059

so now in the right-angled triangle TBR
sin 30 = TR/5940.059
TR = 5940.059*sin30 = 2970.03 ft

Answer 2

d = hor. distance from base of mountain to point where 25o is measured.
Tan25 = h/d, h = d*Tan25.
Tan30 = h//(d-1225), h = (d-1225)*Tan30.

d*Tan25 = (d-1225)*Tan30.
d = (d-1225)*1.23,
d = 6,125 Ft.

h = d*Tan25 = 3064 Ft.