Question
Explain the following text primarily based on the surrounding page content: A bird is perched on the top of a building that is 13 feet tall. You spot the bird as you’re walking across the street. If you are 6 feet tall and the diagonal distance from you to the bird is 26 feet, what is the angle of elevation you are using to look at the bird? Use the inverse of sine to find your answer and round to the nearest whole degree.(1 point) ≈ ≈ °
Answers
GPT-4o mini
To find the angle of elevation from your vantage point to the bird perched on the building, we can use some basic trigonometry, specifically the sine function. The setup involves you standing on the ground, looking up at the bird.
### Given Data:
- Height of the building (where the bird is): 13 feet
- Your height: 6 feet
- Diagonal distance to the bird: 26 feet
### Step-by-Step Calculations:
1. **Determine the height difference**:
The height difference between you and the bird can be calculated as follows:
\[
\text{Height difference} = \text{Height of the building} - \text{Your height} = 13 \text{ feet} - 6 \text{ feet} = 7 \text{ feet}
\]
2. **Visualize the situation**:
When you're looking up at the bird, you form a right triangle where:
- The height difference (7 feet) is the opposite side,
- The diagonal distance (26 feet) is the hypotenuse of the triangle.
3. **Using the sine function**:
The sine of the angle of elevation (let's call it \( \theta \)) is given by the ratio of the opposite side to the hypotenuse:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{7}{26}
\]
4. **Finding the angle \( \theta \)**:
To find the angle \( \theta \), use the arcsine (inverse sine) function:
\[
\theta = \sin^{-1}\left(\frac{7}{26}\right)
\]
5. **Calculating the angle in degrees**:
Using a calculator:
\[
\theta \approx \sin^{-1}(0.2692) \approx 15.7^\circ
\]
6. **Rounding to the nearest whole degree**:
When rounded to the nearest whole degree, \( \theta \approx 16^\circ \).
### Final Answer:
The angle of elevation you are using to look at the bird is approximately **16°**.
### Given Data:
- Height of the building (where the bird is): 13 feet
- Your height: 6 feet
- Diagonal distance to the bird: 26 feet
### Step-by-Step Calculations:
1. **Determine the height difference**:
The height difference between you and the bird can be calculated as follows:
\[
\text{Height difference} = \text{Height of the building} - \text{Your height} = 13 \text{ feet} - 6 \text{ feet} = 7 \text{ feet}
\]
2. **Visualize the situation**:
When you're looking up at the bird, you form a right triangle where:
- The height difference (7 feet) is the opposite side,
- The diagonal distance (26 feet) is the hypotenuse of the triangle.
3. **Using the sine function**:
The sine of the angle of elevation (let's call it \( \theta \)) is given by the ratio of the opposite side to the hypotenuse:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{7}{26}
\]
4. **Finding the angle \( \theta \)**:
To find the angle \( \theta \), use the arcsine (inverse sine) function:
\[
\theta = \sin^{-1}\left(\frac{7}{26}\right)
\]
5. **Calculating the angle in degrees**:
Using a calculator:
\[
\theta \approx \sin^{-1}(0.2692) \approx 15.7^\circ
\]
6. **Rounding to the nearest whole degree**:
When rounded to the nearest whole degree, \( \theta \approx 16^\circ \).
### Final Answer:
The angle of elevation you are using to look at the bird is approximately **16°**.