A person standing 600 ft from the base of a mountain measures the angle of elevation from the ground to the top of the mountain to be 25°. The person then walks 800 ft straight back and measures the angle of elevation to now be 20°. How tall is the mountain?

Answer 1

let the height of the mountain be h ft
let the distance from the base of the mountain to its vertical line h be x ft (inside the mountain)

You have two right-angled triangles.
From the first: tan25 = h/(x+600)
h = (x+600)tan25

From the 2nd: tan 20 = h/(x+1400)
h = (x+1400)tan20

(x+1400)tan20 = (x+600)tan25
xtan20 + 1400tan20 = xtan25 + 600tan25
xtan25 - xtan20 = 1400tan20 - 600tan25
x(tan25-tan20) = 1400tan20-600tan25
x =(1400tan20-600tan25)/(tan25-tan20)
= ... ***
I will let you do the button-pushing, do not round off

then h = (*** + 600)tan25
= .....

This is the traditional way to do this problem.
Let me know if you want a "slicker" way to do this.