Question
An airplane takes off at an angle of elevation of 48°. The airplane travels 976 meters along a diagonal. What is the horizontal distance the plane travels, and what is the plane’s altitude?(1 point)
Responses
The plane travels a horizontal distance of 653.07 m and reaches an altitude of 725.31 m.
The plane travels a horizontal distance of 653.07 m and reaches an altitude of 725.31 m.
The plane travels a horizontal distance of 1,458.61 m and reaches an altitude of 1,083.96 m.
The plane travels a horizontal distance of 1,458.61 m and reaches an altitude of 1,083.96 m.
The plane travels a horizontal distance of 725.31 m and reaches an altitude of 653.07 m.
The plane travels a horizontal distance of 725.31 m and reaches an altitude of 653.07 m.
The plane travels a horizontal distance of 653.07 m and reaches an altitude of 1,174.34 m.
Responses
The plane travels a horizontal distance of 653.07 m and reaches an altitude of 725.31 m.
The plane travels a horizontal distance of 653.07 m and reaches an altitude of 725.31 m.
The plane travels a horizontal distance of 1,458.61 m and reaches an altitude of 1,083.96 m.
The plane travels a horizontal distance of 1,458.61 m and reaches an altitude of 1,083.96 m.
The plane travels a horizontal distance of 725.31 m and reaches an altitude of 653.07 m.
The plane travels a horizontal distance of 725.31 m and reaches an altitude of 653.07 m.
The plane travels a horizontal distance of 653.07 m and reaches an altitude of 1,174.34 m.
Answers
GPT-4o mini
To solve for the horizontal distance and altitude of the airplane, we can use trigonometric functions based on the angle of elevation and the length of the diagonal (which is the hypotenuse of the right triangle formed).
Given:
- Angle of elevation, \( \theta = 48^\circ \)
- Length of the diagonal (hypotenuse), \( h = 976 \, \text{m} \)
1. **Calculating the horizontal distance (adjacent side)**:
\[
\text{Horizontal distance} = h \cdot \cos(\theta)
\]
\[
\text{Horizontal distance} = 976 \cdot \cos(48^\circ) \approx 976 \cdot 0.6691 \approx 653.07 \, \text{m}
\]
2. **Calculating the altitude (opposite side)**:
\[
\text{Altitude} = h \cdot \sin(\theta)
\]
\[
\text{Altitude} = 976 \cdot \sin(48^\circ) \approx 976 \cdot 0.7431 \approx 725.31 \, \text{m}
\]
From the calculations:
- The horizontal distance the airplane travels is approximately **653.07 m**.
- The altitude of the airplane is approximately **725.31 m**.
Therefore, the correct response is:
**The plane travels a horizontal distance of 653.07 m and reaches an altitude of 725.31 m.**
Given:
- Angle of elevation, \( \theta = 48^\circ \)
- Length of the diagonal (hypotenuse), \( h = 976 \, \text{m} \)
1. **Calculating the horizontal distance (adjacent side)**:
\[
\text{Horizontal distance} = h \cdot \cos(\theta)
\]
\[
\text{Horizontal distance} = 976 \cdot \cos(48^\circ) \approx 976 \cdot 0.6691 \approx 653.07 \, \text{m}
\]
2. **Calculating the altitude (opposite side)**:
\[
\text{Altitude} = h \cdot \sin(\theta)
\]
\[
\text{Altitude} = 976 \cdot \sin(48^\circ) \approx 976 \cdot 0.7431 \approx 725.31 \, \text{m}
\]
From the calculations:
- The horizontal distance the airplane travels is approximately **653.07 m**.
- The altitude of the airplane is approximately **725.31 m**.
Therefore, the correct response is:
**The plane travels a horizontal distance of 653.07 m and reaches an altitude of 725.31 m.**