Question

Using trigonometric ratios to solve the following problem. Shaina, who is 5 feet, 6 inches tall, is standing 20 feet from the base of a tree when she sees a pileated woodpecker at the top of the tree the bird is at an angle of elevation of 68 degrees above shaina's line of sight. What is the height of the tree?

Answer 1

To solve for the height of the tree, we can use trigonometric functions, specifically the tangent function, which relates the angle of elevation to the opposite side (the height we want to determine) and the adjacent side (the distance from the tree).

1. **Convert Shaina's height to feet.**
Shaina is 5 feet 6 inches tall. We convert this to feet:
\[
5 \text{ feet} + \frac{6}{12} \text{ feet} = 5.5 \text{ feet}
\]

2. **Identify the given values:**
- Distance from the base of the tree (adjacent side): \( d = 20 \) feet
- Angle of elevation: \( \theta = 68^\circ \)
- Shaina's height: \( h_s = 5.5 \) feet

3. **Use the tangent function:**
The height of the tree above Shaina's height can be found using the tangent of the angle of elevation:
\[
\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h - h_s}{d}
\]

Rearranging for \( h \) (the height of the tree):
\[
h - h_s = d \cdot \tan(\theta)
\]
\[
h = h_s + d \cdot \tan(\theta)
\]

4. **Calculate \( \tan(68^\circ) \):**
Using a calculator, we find:
\[
\tan(68^\circ) \approx 2.4751
\]

5. **Calculate the height of the tree:**
Substitute the values into the height formula.
\[
h = 5.5 + 20 \cdot 2.4751
\]
\[
h = 5.5 + 49.502 = 55.002
\]

Therefore, the approximate height of the tree is:
\[
\text{Height of the tree} \approx 55.0 \text{ feet}
\]

**Final Answer:**
The height of the tree is approximately **55 feet**.