Question

Corey spies a bald eagle in a tall tree. He estimates the height of the tree to be 24 feet and the angle of elevation to the bird from where he stands to be 52°. 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 33°. Round answers to the nearest hundredth of a foot. How far is Corey from the tree in the initial position? How far is Corey from the tree in the new position? How many feet did Corey have to move back to get to the new position?

Answer 1

Corey was initially about 19.12 feet from the tree. After repositioning, he estimated a distance of approximately 31.92 feet. Corey moved approximately 12.80 feet to attain a better view.

Let's denote the initial distance from Corey to the tree as
`\(d_1\)`, the new distance as
`\(d_2\)`, and the height of the tree as h.

Using trigonometry, we can set up two right triangles to solve for
`\(d_1\)` and
`\(d_2\)`.

1. Initial Position:

-
`\(d_1 = \frac{h}{\tan(\text{angle of elevation}_1)}\)`

-
`\(d_1 = (24)/(\tan(52^\circ)) \approx 19.12\)` feet

2. New Position:

-
`\(d_2 = \frac{h}{\tan(\text{angle of elevation}_2)}\)`

-
`\(d_2 = (24)/(\tan(33^\circ)) \approx 31.92\)` feet

3. Distance Moved:

-
`\( \text{Distance Moved} = d_2 - d_1 \)`

-
`\( \text{Distance Moved} \approx 31.92 - 19.12 \approx 12.80\)` feet

So, Corey was initially approximately 19.12 feet from the tree, moved to a new position approximately 31.92 feet from the tree, and had to move back around 12.80 feet to get the better view.

Answer 2

Corey is initially approximately 29.16 feet from the tree, in the new position he is approximately 42.84 feet from the tree, and he moved back approximately 13.68 feet to get to the new position.

In the initial position, Corey estimates the height of the tree to be 24 feet, and the angle of elevation to the bird is measured at 52°. By using trigonometric ratios, specifically the tangent function, Corey can determine his initial distance from the tree. The formula tan(angle)=height/distance can be rearranged to find the distance, resulting in Corey being approximately 29.16 feet away from the tree.

Upon stepping back to get a better view, Corey estimates a new angle of elevation to be 33°. Using the same trigonometric approach, Corey can calculate the new distance from the tree. The new position places Corey approximately 42.84 feet away from the tree.

To find how far Corey moved back, we subtract the initial distance from the new distance, yielding approximately 13.68 feet. Therefore, Corey had to move back approximately 13.68 feet to achieve the new position and get a better view of the bald eagle. The trigonometric principles employed in this problem showcase the practical application of geometry in estimating distances and angles in real-world scenarios.