https://file.aiquickdraw.com/mathbot/file/17264544188754cixmbqe.jpg

The problem involves an airplane flying at an altitude of 6.5 miles, with the task being to find the horizontal distance \( d \) from the observer to the airplane for different values of the angle of elevation \( \theta \).

This is a classic trigonometry problem that uses the right triangle formed by the altitude, the distance \( d \), and the line from the observer to the plane. The relationship is given by:

\[
\text{tan}(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{6.5}{d}
\]

Solving for \( d \):

\[
d = \frac{6.5}{\text{tan}(\theta)}
\]

### For each part:

1. **(a) \( \theta = 30^\circ \):**
   \[
   d = \frac{6.5}{\tan(30^\circ)} = \frac{6.5}{0.577} \approx 11.3 \, \text{miles}
   \]

2. **(b) \( \theta = 90^\circ \):**
   At \( \theta = 90^\circ \), the plane is directly above the observer, so:
   \[
   d = 0 \, \text{miles}
   \]

3. **(c) \( \theta = 120^\circ \):**
   \( \theta \) is now an obtuse angle, so the situation changes slightly. However, the horizontal distance doesn't make sense to calculate for angles greater than 90° in this context, as the plane would have already passed the observer. If needed, you can use the cosine law or rework the geometry.

Let me know if you want more details on any part!

---

Here are some related questions:

1. How does the tangent function relate to right triangles?
2. What happens to the distance \( d \) as \( \theta \) approaches \( 90^\circ \)?
3. How would the distance formula change if the altitude of the airplane increased?
4. How can the problem be solved using different trigonometric functions?
5. What if the airplane was flying on a curved path—how would this change the trigonometric relationship?

**Tip:** Always make sure to set your calculator to the correct mode (degrees/radians) when solving trigonometric problems.

An airplane, flying at an altitude of 6.5 miles, is on a flight path that passes directly over an observer. Find the distance (in miles) from the observer to the plane for each value of the angle of elevation (θ). Values of θ: (a) 30°, (b) 90°, and (c) 120°.

This trigonometry problem helps you learn how to calculate the distance from an observer to an airplane flying at an altitude of 6.5 miles using the tangent function. It includes solutions for angles of 30°, 90°, and 120°.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation