# How do you find the height of a tree if the angle of elevation of its top changes from 20 degrees to 40 degrees as the observer advances 23 meters towards the base?

Nov 11, 2015

7.61 meters

#### Explanation:

In this problem we are essentially given two triangles with 2 known angles, one known side, and one assumed (90 degree triangle). The first triangle that we will use will be the smaller one.

We're told that the angle of elevation is 20 degrees and then 40 degrees when we move in 23 meters. this gives us enough information. It is our goal to gather the required information to solve the triangle with the "tree" that we are trying to find the height of. Therefore we don't need to solve the entire triangle.

Using the Law of Sines $\frac{a}{S} \in A$=$\frac{b}{S} \in B$=$\frac{c}{S} \in C$

We can find the hypotenuse of the right angle.

$\frac{a}{S} \in 20 = \frac{23}{S} \in 40$ which gives us the length of the hypotenuse as:

Through manipulation $\frac{23 S \in 20}{S} \in 40$ = $12.23 m$

Using the Law of Sines once more to solve for the right triangle, our equation looks like:

$\frac{12.23}{S} \in 90$=$\frac{a}{S \in 40}$

Through manipulation $\frac{12.23 S \in 40}{S} \in 90$ = $7.86 m$

The important thing about this kind of problem is being able to recognize what you are given and what you need and deductive reasoning to deduce the figures you need.