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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.) 

This problem involves trigonometry and can be solved using the law of sines to find the height \( h \) of the tree.

### Step-by-step process:

1. **Labeling the triangle**:
   - \( A \) and \( B \) are the two points on the ground.
   - The angle of elevation from point \( A \) to the top of the tree is \( 8.4^\circ \).
   - The distance between points \( A \) and \( B \) is \( 24.8 \) feet.
   - The angle between \( AB \) and the line from \( B \) to the top of the tree is \( 87.6^\circ \).

2. **Triangle Angles**:
   We need to find the third angle in the triangle (let's call it \( C \), the angle at the tree's base):
   \[
   \text{Angle } C = 180^\circ - 87.6^\circ - 8.4^\circ = 84^\circ
   \]

3. **Using the Law of Sines**:
   We can apply the Law of Sines to find the distance from \( B \) to the base of the tree (let's call it \( d \)).

   \[
   \frac{d}{\sin(8.4^\circ)} = \frac{24.8}{\sin(84^\circ)}
   \]

4. **Finding the height**:
   Once we have the distance \( d \), we can find the height \( h \) of the tree using basic trigonometry, specifically the sine function from point \( A \):

   \[
   h = d \cdot \sin(8.4^\circ)
   \]

Let's calculate this step-by-step.The height of the tree is approximately 0.5 feet, which seems too small given the context. It looks like we misinterpreted a step. Let's revisit the equation and correctly apply the law of sines:

We need to use the law of sines to find the full distance to the tree, and then recalculate the height. I'll adjust the calculation accordingly.It seems like I can’t do more advanced data analysis right now. Please try again later. 

However, I can explain the method and process again if you'd like to try calculating it manually. Would you like the steps or a breakdown of the calculation process?

Here are five related questions you could explore:
1. How is the law of sines applied to triangles with oblique angles?
2. How can we calculate the height of an object using angles and distances?
3. What is the difference between the law of sines and the law of cosines?
4. How would the approach change if the distances were not measured perpendicularly?
5. How does the angle of elevation affect the calculation of height?

**Tip**: When using the law of sines, make sure to double-check the angle values for accurate trigonometric calculations.

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.)

Discover how to calculate the height of a redwood tree using trigonometry and the Law of Sines, starting from basic angles of elevation and distances. This guide is perfect for high school students looking to apply real-world scenarios to geometry concepts.

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