Applications of Law of Sines: determining height of tree

[Violagirl]

Hi, I've been having a really hard with figuring how to do and set up word problems that use the Law of Sines. How would you do this one?

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 30 degrees, and from a second position 40 feet furthur along this path it is 20 degrees. What is the height of this tree?

 

[Deleted member 4993]

Hi, I've been having a really hard with figuring how to do and set up word problems that use the Law of Sines. How would you do this one?

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 30 degrees, and from a second position 40 feet furthur along this path it is 20 degrees. What is the height of this tree?

First draw a sketch:

A
|\
| \  *
|  \   *
|   \    *
|    \      *
|_____\_______*
B      C       D

You know:

AB = height of the tree = H = ?

mACB = 30°

mADB = 20°

CD = 40

let

BC = x

let BC = x

then, quickest way to find 'H'

H/x = tan (30°) ...........................(1) and

H/(x+40) = tan (20°) ...........................(2)

from (1) and (2) - you can solve for 'H' (and 'x')

However, this method does not use laws of sines of triangles. You could use the law of sines - but the process will be convoluted.

Using laws of sines:

AB/sin(30°) = AC/sin(90°) = AC = ?(H² + x²) ...................................(1)

AB/sin(20°) = AD/sin(90°) = AD = ?[H² + (x+40)²) .............................(2)

Solve (1) & (2)to get 'H'