Question
Use 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?
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Responses
Answers
GPT-4o mini
To find the height of the tree, we can use trigonometric ratios, specifically the tangent function which relates the angle of elevation to the opposite side (height of the tree above Shaina's eyes) and the adjacent side (distance from Shaina to the base of the tree).
1. **Convert Shaina's height to feet**: Shaina is 5 feet 6 inches tall. We convert inches to feet:
\[
6 \text{ inches} = \frac{6}{12} \text{ feet} = 0.5 \text{ feet}
\]
Therefore, Shaina's total height is:
\[
5 + 0.5 = 5.5 \text{ feet}
\]
2. **Identify the components of the situation**:
- The distance from Shaina to the base of the tree: \( 20 \) feet (adjacent side)
- The angle of elevation to the bird: \( 68^\circ \)
3. **Use the tangent function**:
The tangent of the angle of elevation is given by:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
In this case, the opposite side is the height of the tree above Shaina's eyes, and the adjacent side is the distance from Shaina to the base of the tree.
Let \( h \) be the height above Shaina's eyes. Then,
\[
\tan(68^\circ) = \frac{h}{20}
\]
4. **Solve for \( h \)**:
Rearranging gives:
\[
h = 20 \cdot \tan(68^\circ)
\]
Now, we can calculate \( \tan(68^\circ) \) using a calculator:
\[
\tan(68^\circ) \approx 2.4751
\]
Thus,
\[
h \approx 20 \cdot 2.4751 \approx 49.502 \text{ feet}
\]
5. **Calculate the total height of the tree**:
The total height of the tree is the height above Shaina’s eyes plus Shaina’s height:
\[
\text{Total Height} = h + 5.5 \approx 49.502 + 5.5 \approx 55.002 \text{ feet}
\]
So, the height of the tree is approximately **55 feet**.
1. **Convert Shaina's height to feet**: Shaina is 5 feet 6 inches tall. We convert inches to feet:
\[
6 \text{ inches} = \frac{6}{12} \text{ feet} = 0.5 \text{ feet}
\]
Therefore, Shaina's total height is:
\[
5 + 0.5 = 5.5 \text{ feet}
\]
2. **Identify the components of the situation**:
- The distance from Shaina to the base of the tree: \( 20 \) feet (adjacent side)
- The angle of elevation to the bird: \( 68^\circ \)
3. **Use the tangent function**:
The tangent of the angle of elevation is given by:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
In this case, the opposite side is the height of the tree above Shaina's eyes, and the adjacent side is the distance from Shaina to the base of the tree.
Let \( h \) be the height above Shaina's eyes. Then,
\[
\tan(68^\circ) = \frac{h}{20}
\]
4. **Solve for \( h \)**:
Rearranging gives:
\[
h = 20 \cdot \tan(68^\circ)
\]
Now, we can calculate \( \tan(68^\circ) \) using a calculator:
\[
\tan(68^\circ) \approx 2.4751
\]
Thus,
\[
h \approx 20 \cdot 2.4751 \approx 49.502 \text{ feet}
\]
5. **Calculate the total height of the tree**:
The total height of the tree is the height above Shaina’s eyes plus Shaina’s height:
\[
\text{Total Height} = h + 5.5 \approx 49.502 + 5.5 \approx 55.002 \text{ feet}
\]
So, the height of the tree is approximately **55 feet**.