Height of a tree, given angle of elevation and....

[Timcago]

You want to know the height of the tree in the figure above. From point A, you find that the angle of elevation to the top of the tree is 10.7 degrees. You then move 24.8 feet at a right angle from point A. You find the angle from the path you just walked to the base of the tree (angle B) is 86.6 degrees. How tall is the tree?


This is really hard, because it only gives 1 angle for 2 of the triangles and only one leg length that works for 2 triangles.

If there is a right triangle in here then it will be a completely different problem, but i do not understand what the problem means when it says "You then move 24.8 feet at a right angle from point A."

Does this mean that Angle A is 90 degrees?

Can anyone figure this one out?

 

[skeeter]

let point C be at the base of the tree ...
ABC is a right triangle

|AC|/24.8 = tan(86.6)
|AC| = 24.8tan(86.6)

let point T be the top of the tree ...
ACT is a right triangle

|CT|/|AC| = tan(10.7)

height of the tree, |CT| = |AC|tan(10.7) = 24.8tan(86.6)tan(10.7) = 78.9 ft

 

[soroban]

Re: Height of a tree

Hello, Timcago!

Let me type some diagrams to accompany skeeter's excellent solution.

      C
      *
      |\
      | \        (Looking down at the ground)
      |  \
      |   \
      |    \
      |     \
      |      \
      |       \
      |   86.6°\
      * - - - - * 
      A   24.8  B

Let point \(\displaystyle C\) be at the base of the tree.
\(\displaystyle ABC\) is a right triangle

\(\displaystyle \tan86.6^o\:=\:\L\frac{AC}{24.8}\)\(\displaystyle \;\;\Rightarrow\;\;AC \:= \:24.8(\tan86.6^o)\:\approx\:417.43\)

      T
      *
      | \
      |   \       (side view)
      |     \
      |       \
      |         \
      |           \
      |       10.7° \
      * - - - - - - - *
      C    417.43   A

Let point \(\displaystyle T\) be the top of the tree.
\(\displaystyle ACT\) is a right triangle

\(\displaystyle \tan(10.7) \:= \:\L\frac{CT}{417.43}\)

Height of the tree: \(\displaystyle \,CT \:= \:417.43(\tan10.7^o) \;\approx\;78.9\) ft
.