Question

the top of a building 24m high is observed from the top of and from the bottom of a vertical tree. The angles of elevation are found to be45 and 60 degrees respectively. Find the height of the tree leaving your answer in surds form and, in a diagram

Answer 1

Let the tree height be h (m) and the horizontal distance between tree and building be x (m).

Sketch (side view, not to scale):

building (height 24)
|\
| \ top of building
24| \
| \
| \ angle 60° at base of tree
| \
| \
| \
| \
| \
| \
| \ top of tree (height h) — angle 45° to top of building
| \
----------------- ground
x

From the top of the tree: tan 45° = (24 − h)/x = 1, so
24 − h = x. (1)

From the base of the tree: tan 60° = 24/x = √3, so
x = 24/√3 = 8√3. (2)

Substitute (2) into (1):
h = 24 − x = 24 − 8√3 = 8(3 − √3) metres.

(Approximately h ≈ 10.144 m.)