Question

Corey spies a bald eagle in a tall tree. He estimates the height of the tree to be 80 feet and the angle of elevation to the bird from where he stands to be 68°. The leaves on the tree make it difficult for Corey to watch the bird, so he takes several steps away from the tree to get a better view. He now estimates his angle of elevation to be 41°.
How many feet did Corey step back to gain a better view of the bird? Round your answer to the nearest hundredth of a foot.

Answer 1

Answer:

Corey stepped back 59.71 feet

Step-by-step explanation:

Right Triangles

They are a special type of triangle where one of its internal angles is 90°. The basic trigonometric equations stand in this type of triangles. If x is the adjacent leg of a given angle $\theta$, and y is the opposite leg to the same angle, then :

$\displaystyle tan\theta=\frac{y}{x}$

We can solve for x to get

$\displaystyle x=\frac{y}{tan\theta}$

Corey estimates the height of the tree to be 80 feet. This means we know y=80 and $\theta=68^o$. Let's find the horizontal distance at which Corey is looking at the bald eagle:

$\displaystyle x_1=\frac{80}{tan68^o}$

$x_1=32.32\ feet$

Now Corey moves back to watch the very same tree at an elevation angle of 41°. The tree has the same height, so

$\displaystyle x_2=\frac{80}{tan41^o}$

$x_2=92.03\ feet$

Now we can know the distance Corey walked back by subtracting both distances

$x=x_2-x_1=92.03-32.32=59.71\ feet$

$\boxed{\text{Corey stepped back 59.71 feet}}$