Say I'm at an unknown distance from a mountain, called point P, and I estimate the angle of elevation to the top of the mountain is 13.5 degrees. Then I move to point N, which is 100 meters closer to the mountain, and I estimate the angle of elevation to be 14.8 degrees. What is the height of the mountain?

review the basic trig functions. You will see that if the height is h,

h cot13.5° - h cot 14.8° = 100

Tan13.5 = h/P, h = P*Tan13.5.

Tan14.8 = h/(P-100), h = (P-100)*Tan14.8.

P*Tan13.5 = (P-100)*Tan14.8,
P = (P-100)*1.10,
0.91P = P-100,
P = 1,111 Ft.

h = P*Tan13.5 = 1111*Tan13.5 = 266.8 Ft.

To solve this problem, we can use trigonometry principles and create a system of equations.

Let's assume the height of the mountain is h meters and the distance from point P to the mountain is d meters. We can then create the following equations:

Equation 1: tan(13.5°) = h / d
Equation 2: tan(14.8°) = h / (d - 100)

To find the height of the mountain, we need to solve for h. Let's start by simplifying the equations:

Equation 1 becomes: h = d * tan(13.5°)
Equation 2 becomes: h = (d - 100) * tan(14.8°)

Now we have two equations with h on both sides. We can equate the right sides of the equations and solve for d:

d * tan(13.5°) = (d - 100) * tan(14.8°)

To solve this equation, we can isolate the d term:

d * tan(13.5°) - d * tan(14.8°) = -100 * tan(14.8°)

Factor out d:

d * (tan(13.5°) - tan(14.8°)) = -100 * tan(14.8°)

Now we can solve for d:

d = (-100 * tan(14.8°)) / (tan(13.5°) - tan(14.8°))

Once we have the value of d, we can substitute it into either of the original equations (Equation 1 or Equation 2) to find the height of the mountain (h).