SOLUTION: In traveling across flat land, you notice a mountain directly in front of you. Its angle of elevation (to the peak) is 3.5 o . After you drive 13 miles closer to the mountain, the

Algebra ->  Trigonometry-basics -> SOLUTION: In traveling across flat land, you notice a mountain directly in front of you. Its angle of elevation (to the peak) is 3.5 o . After you drive 13 miles closer to the mountain, the      Log On


   

Question 1197746: In traveling across flat land, you notice a mountain directly in front of you. Its angle of elevation
(to the peak) is 3.5 o . After you drive 13 miles closer to the mountain, the angle of elevation is 9 o .
Approximate the height of the mountain.
Found 2 solutions by josgarithmetic, math_tutor2020:
Answer by josgarithmetic(39618) About Me  (Show Source):
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

This is one way to draw the diagram.

A = 1st observation point
B = 2nd observation point
C = base of the mountain
D = peak of the mountain

The goal is to determine the length of segment CD, aka the height of the mountain, which I'll call h.
Another variable we'll use is x which is the distance from B to C.

Focus your attention on triangle BCD.
Use the tangent ratio to say this
tan(angle) = opp/adj
tan(B) = CD/BC
tan(9) = h/x
x*tan(9) = h
x = h/tan(9)
x = (1/tan(9))*h
x = 6.3137515h
This is approximate.
Your calculator needs to be in degree mode.

Now move your focus to triangle ACD.
tan(angle) = opp/adj
tan(A) = CD/AC
tan(3.5) = h/(x+13)
tan(3.5) = h/(6.3137515h+13)
0.0611626 = h/(6.3137515h+13)

Let's solve for h.
0.0611626 = h/(6.3137515h+13)
0.0611626(6.3137515h+13) = h
0.0611626*6.3137515h+0.0611626*13 = h
0.3861655h+0.7951138 = h
0.7951138 = h-0.3861655h
0.7951138 = 0.6138345h
0.6138345h = 0.7951138
h = 0.7951138/0.6138345
h = 1.29532276208
h = 1.2953

Answer: Approximately 1.2953 miles tall

Extra info:
Multiply this with 5280 to convert to feet
1.2953 miles = 5280*1.2953 = 6839.184 feet