Question

Pat needs to determine the height of a tree before cutting it down to be sure that it will not fall on a nearby fence. the angle of elevation of the tree from one position on a flat path from the tree is upper h equals 40 degrees comma h=40°, and from a second position upper l equals 60 feet l=60 feet farther along this path it is upper b equals 30 degrees . b=30°. what is the height of the​ tree?

Answer 1

Let the distance between the tree and the first point (point H) be x and let the height of the tree be h, then:

$\tan40^o= \frac{h}{x} \\ \\ \Rightarrow h=x\tan40^o \ . \ . \ . \ (1)$

Also, the distance from the tree to the second point (point L) is x + 60, thus:

$\tan30^o= \frac{h}{x+60} \\ \\ \Rightarrow h=(x+60)\tan30^o \ . \ . \ . \ (2)$

From (1) and (2), we have:

$x\tan40^o=(x+60)\tan30^o \\ \\ \Rightarrow 0.8391x=0.5774x+34.64 \\ \\ \Rightarrow0.2617x=34.64 \\ \\ \Rightarrow x= \frac{34.64}{0.2617} =132.4 \ feet$

From (1):

h = x tan 40° = 132.4 (0.8391) = 111.1 feet.

Therefore, the height of the tree is 111.1 feet