In this question, we discover the height of a mountain which is surrounded by rugged terrain.

Question

In this question, we discover the height of a mountain which is surrounded by rugged terrain. Suppose A and B are two towns located nearby a mountain whose base is at M and whose peak is at N. The citizens want to measure the height of the mountain. Unfortunately, they are not able to get close enough to the base of the mountain to measure the distances (line) BM or (line) AM We have the following facts:

1. Town A is located exactly east of the mountain.
2. Town B is located exactly southwest of the mountain.
3. The angles of elevation to the peaks of the mountain are 13∘  and 8∘ as shown.
4. The towns are 20 miles apart.

Let h=(line)MN be the height of the mountain, and let a and b be the distances to the towns A and B respectively.

1) Find an expression for a in terms of h: 

2) Similarly, find b in terms of h: 

3) Find the value of ∠BMA:

4) Finally, use the cosine rule on the triangle ΔBMA and solve for h to obtain the height of the mountain in miles:

Answer 1

Hi Adri,

Use:

a = h/tan(13)

b = h/tan(8)

20² = a² + b² - 2(a)(b)cos(135)

h = 1.88 mi

If you need help setting up the triangles let me know.

I hope this helps, Joe.

Answer 2

You do not specify which town has 8° and which has 12°.

I have given 8° to town A.

1)

tan 8º = h / a

a tan 8º = h

a = h / tan 8º

2)

repeat process for b

3)

135º

4)

h is not included in ΔBMA. Review post for accuracy and/or missing information.