Question

Two people are 100 feet apart on opposite sides of a tree. The angles of elevation from the people to the top of the tree are 30° and 45°. Find the height of the tree. Round your answer to the nearest tenth of a foot.

Answer 1

36.6 ft

Explanation:

First, find the angle at the top of the tree between the people: 180° - 30° - 45° = 105°.

Next, use the Law of Sines:

a

sin A

=

b

sin B

=

c

sin C

to find the distance from the top of the tree to the second person.

x

sin (30°)

=

100 ft

sin (105°)

x ≈ 55.1689 ft

Now, you can find the height of the tree by using a 45-45-90 special right triangle, so the height =

55.1689

2

≈ 36.6 ft

Answer 2

We need to solve for the height of the tree given two angles and distance between the two observers. See attached drawing for a better understanding of the problem.
We derive to equations using SOH CAH TOA such as below:
sin30 = h / x
sin 45 = h / (100-x)

sin 45 (100-x) = xsin30
70.71 - 0.71x = 0.5x
70.71 = 1.21 x
x = 58.44

Solving for h, we have:
h = xsin30
h = 58.44 sin30
h = 29.22

The height of the tree is 29.22 feet.