Trigonometric word problem involving the height of a tree using the angle of elevation.

Problem Analysis:

We are given a trigonometric word problem involving the height of a tree. The key details are:
- The distance from the point on the ground to the foot of the tree is 47 feet.
- The angle of elevation from this point to the top of the tree is \(35^\circ\).
- We need to find the height of the tree.

Diagram Interpretation:

The diagram provided shows:
- Point \( A \) is the observer's position on the ground, 47 feet away from the base of the tree.
- Point \( C \) is the base (foot) of the tree.
- Point \( B \) is the top of the tree.
- The angle of elevation from \( A \) to \( B \) is \(35^\circ\).
- The height of the tree is represented by the vertical line segment \( BC \), which we need to find.

Solution Approach:

This is a classic right triangle problem where:
- The distance from the observer to the base of the tree (\( AC \)) is the adjacent side.
- The height of the tree (\( BC \)) is the opposite side.
- The angle of elevation (\( \angle BAC \)) is \(35^\circ\).

We can use the tangent function in trigonometry, which relates the opposite side to the adjacent side in a right triangle:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Here:
- \(\theta = 35^\circ\)
- Opposite side = \( BC \) (height of the tree)
- Adjacent side = \( AC = 47 \) feet

Thus:
\[
\tan(35^\circ) = \frac{BC}{47}
\]

Step-by-Step Calculation:

1. Express the equation:
\[
\tan(35^\circ) = \frac{BC}{47}
\]

2. Solve for \( BC \):
\[
BC = 47 \cdot \tan(35^\circ)
\]

3. Use a calculator to find \(\tan(35^\circ)\):
\[
\tan(35^\circ) \approx 0.7002
\]

4. Substitute the value into the equation:
\[
BC = 47 \cdot 0.7002
\]

5. Perform the multiplication:
\[
BC \approx 32.91
\]

Final Answer:

The height of the tree is approximately \(32.91\) feet.

\[
\boxed{32.91}
\]