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.