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.

How high (in feet) is the mountain?

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

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.

To find the height of the mountain, we can use trigonometry and create a right triangle.

Let's label the height of the mountain as "h" (in feet) and the distance from the first observation point to the base of the mountain as "x" (in feet).

From the first observation point, we can create a right triangle with the height of the mountain as the vertical leg and the horizontal leg as "x". The angle opposite the height of the mountain is 25 degrees.

Similarly, from the second observation point, which is 1225 feet closer to the mountain along the plain, we can create another right triangle. The height of the mountain is still the vertical leg, and the horizontal leg is now "x - 1225". The angle opposite the height of the mountain in this case is 30 degrees.

Now, we can use trigonometry to create two equations.

In the first right triangle:
tan(25) = h / x

In the second right triangle:
tan(30) = h / (x - 1225)

We can solve these equations to find the values of h and x.

First, let's solve the first equation for x:
x = h / tan(25)

Next, substitute this value of x into the second equation:
tan(30) = h / ((h / tan(25)) - 1225)

Now, we can solve for h.

To do this, first, multiply both sides of the equation by ((h / tan(25)) - 1225):
tan(30) * ((h / tan(25)) - 1225) = h

Next, distribute and combine like terms:
(h / tan(25)) * tan(30) - 1225 * tan(30) = h

Now, multiply (h / tan(25)) * tan(30):
h * tan(30) - 1225 * tan(30) = h

Finally, isolate the variable h:
h * (tan(30) - 1) = 1225 * tan(30)
h = (1225 * tan(30)) / (tan(30) - 1)

By calculating this expression, we can find the height of the mountain.