https://file.aiquickdraw.com/mathbot/file/1728919142633u0mx6mos.jpg

Let's break down the problem step by step.

### Problem Setup:
- The tree grows at an angle of \( 2^\circ \) from the vertical due to wind.
- The horizontal distance from the base of the tree is \( d = 35 \, \text{m} \).
- The angle of elevation from the point on the ground to the top of the tree is \( a = 25^\circ \).

### Part (a):
You are asked to write an equation using the Law of Sines, which is suitable since we are working with angles and sides of a triangle.

We have two angles:
1. The angle between the tree (which is vertical) and the top is \( b = 90^\circ - 2^\circ = 88^\circ \) (because the tree grows at an angle of \( 2^\circ \) from vertical).
2. The angle of elevation from the point on the ground is \( a = 25^\circ \).

Using the Law of Sines:
\[
\frac{h}{\sin(a)} = \frac{35}{\sin(b)}
\]
Where:
- \( h \) is the height of the tree (the side opposite angle \( a \)).
- \( 35 \, \text{m} \) is the distance from the tree (the side opposite angle \( b \)).
- \( a = 25^\circ \)
- \( b = 88^\circ \)

So, the equation is:
\[
\frac{h}{\sin(25^\circ)} = \frac{35}{\sin(88^\circ)}
\]

### Part (b):
To find \( h \), rearrange the equation:
\[
h = \frac{35 \times \sin(25^\circ)}{\sin(88^\circ)}
\]

Let's calculate this:

- \( \sin(25^\circ) \approx 0.4226 \)
- \( \sin(88^\circ) \approx 0.9994 \)

Substitute these values:
\[
h = \frac{35 \times 0.4226}{0.9994} \approx 14.79 \, \text{m}
\]

### Final Answer:
The height \( h \) of the tree is approximately \( 14.8 \, \text{m} \) (rounded to one decimal place).

---

Would you like further details on the solution? Here are five related questions for deeper understanding:

1. How does the Law of Sines apply to non-right triangles?
2. How would the equation change if the angle of elevation was different?
3. What is the importance of converting the tree's tilt into angle \( b \)?
4. How does rounding affect the accuracy of this solution?
5. Can this method be applied to other real-world applications, like measuring building heights?

**Tip**: Remember to double-check your trigonometric values when solving angle-related problems for precision.

A tree grows at an angle of 2° from the vertical due to prevailing winds. At a point d = 35 meters from the base of the tree, the angle of elevation to the top of the tree is a = 25°. Write an equation using the Law of Sines to find the height h of the tree, and calculate h.

Learn how to calculate the height of a tree growing at an angle of 2° from vertical. Using a distance of 35 meters from the tree base and an angle of elevation of 25°, apply the Law of Sines to find the tree height. Suitable for high school students in Grades 9-12.

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