This problem involves applying trigonometry, specifically the law of sines, to find the length of the tree.

From the image, we know:

• The distance from the person to the tree's base is 95 feet.
• The angle of elevation from the person to the top of the tree is $37^\circ$.
• The tree is growing at an angle of $86^\circ$ to the ground.

We are tasked with finding the length $x$, the tree's height.

Solution:

The triangle involves three sides and angles, and we can use the Law of Sines, which is given by:

$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

Where:

• $A$, $B$, and $C$ are the angles of the triangle,
• $a$, $b$, and $c$ are the sides opposite those angles.

In this case:

• $A = 37^\circ$ (angle of elevation),
• $B = 86^\circ$ (tree's tilt angle),
• $C = 180^\circ - (37^\circ + 86^\circ) = 57^\circ$ (remaining angle).

Now applying the Law of Sines:

$\frac{x}{\sin(37^\circ)} = \frac{95}{\sin(57^\circ)}$

Solving for $x$:

$x = \frac{95 \times \sin(37^\circ)}{\sin(57^\circ)}$

Now calculate the sine values and solve:

$x = \frac{95 \times 0.6018}{0.8387} \approx \frac{57.17}{0.8387} \approx 68.16 \, \text{feet}$

So, the height of the tree is approximately 68.2 feet (rounded to the nearest tenth).

Would you like further details or clarifications on any part of this process?

Here are some related questions:

1. What is the Law of Sines, and how is it applied in different triangle types?
2. How do you handle cases where you cannot use the Law of Sines directly?
3. What is the importance of significant figures in real-life calculations like this?
4. How does the angle of elevation help in finding heights or distances?
5. Can the Law of Cosines be used in this problem, and if so, how?

Tip: Double-check your angle sums when applying the Law of Sines or any triangle-based calculation to avoid mistakes.