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An ecologist wishes to find the height of a redwood tree that is on the other side of a creek, as shown in the figure below. From point *A* he finds that the angle of elevation to the top of the tree is 8.4°. He then walks 24.8 feet at a right angle from point *A* to point *B*. There he finds that the angle between *AB* and a line extending from *B* to the tree is 87.6°. What is the height *h* of the tree? (Round your answer to one decimal place.) 

To determine the height \( h \) of the tree, we can solve this problem using trigonometry. Here's the step-by-step process:

### Given:
- Angle of elevation from point \( A \) to the top of the tree: \( \theta_1 = 8.4^\circ \).
- Distance between point \( A \) and \( B \): \( AB = 24.8 \) feet.
- Angle between line \( AB \) and the line from point \( B \) to the tree: \( \theta_2 = 87.6^\circ \).

We can solve the problem using the Law of Sines and trigonometry:

### Step 1: Set up the Law of Sines
We are looking for the height \( h \) of the tree. We need to find the distance from point \( A \) to the base of the tree, which we will call \( x \).

Since we have an oblique triangle formed by points \( A \), \( B \), and the tree, we can use the Law of Sines to find \( x \). From the triangle \( ABT \) (with \( T \) being the tree base):

\[
\frac{x}{\sin(87.6^\circ)} = \frac{24.8}{\sin(8.4^\circ)}
\]

Let me calculate \( x \) first.

It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can still explain the next steps or walk you through solving the problem manually.

Would you like me to continue with an explanation or assist you in a different way? 

Here are some related questions for further exploration:
1. How can you apply the Law of Sines in other real-world scenarios?
2. How does changing the angles affect the height calculation?
3. What are the limitations of using trigonometry for height estimation in real-life settings?
4. How can you measure angles and distances more accurately in a field environment?
5. Can you use a different method, like the Law of Cosines, to solve the same problem?

**Tip:** Always ensure the angles in your triangle add up to 180° in oblique triangle problems.

An ecologist wishes to find the height of a redwood tree that is on the other side of a creek. From point A he finds that the angle of elevation to the top of the tree is 8.4°. He then walks 24.8 feet at a right angle from point A to point B. There he finds that the angle between AB and a line extending from B to the tree is 87.6°. What is the height h of the tree? (Round your answer to one decimal place.)

This problem involves calculating the height of a redwood tree using trigonometry and the Law of Sines. Given two angles of elevation and the distance between two points, this step-by-step solution explains how to solve for the height of the tree. Suitable for high school students in Grades 9-12.

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