A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 9 mi apart, to be ∠*x* = 27° and ∠*y* = 53°, as shown in the figure. (Round your answers to two decimal places.) Find the elevation of the plane.

We can solve this problem by modeling it using trigonometry, particularly right triangles.

### Step-by-step approach:

1. **Identify what is given:**
   - Two angles of depression: ∠*x* = 27° and ∠*y* = 53°.
   - The distance between the two mileposts is 9 miles.

2. **Set up the right triangles:**
   - Let the elevation of the plane be \( h \).
   - Let the horizontal distances from the plane's vertical projection point to the closer and farther mileposts be \( d_1 \) and \( d_2 \), respectively. So, we have \( d_2 - d_1 = 9 \) (the distance between the two mileposts).

3. **Use the tangent function:**
   Since the angles of depression form right triangles, we can use the tangent function, which relates the opposite side (the elevation, \( h \)) to the adjacent side (the horizontal distance, \( d_1 \) or \( d_2 \)).

   - For the first triangle (with angle 53°), the relation is:
     \[
     \tan(53^\circ) = \frac{h}{d_1}
     \]
     So, 
     \[
     d_1 = \frac{h}{\tan(53^\circ)}
     \]

   - For the second triangle (with angle 27°), the relation is:
     \[
     \tan(27^\circ) = \frac{h}{d_2}
     \]
     So,
     \[
     d_2 = \frac{h}{\tan(27^\circ)}
     \]

4. **Set up an equation:**
   Since the distance between the mileposts is 9 miles, we have:
   \[
   d_2 - d_1 = 9
   \]
   Substituting the expressions for \( d_1 \) and \( d_2 \) from the tangent equations:
   \[
   \frac{h}{\tan(27^\circ)} - \frac{h}{\tan(53^\circ)} = 9
   \]

5. **Solve for \( h \):**
   Factor out \( h \):
   \[
   h \left( \frac{1}{\tan(27^\circ)} - \frac{1}{\tan(53^\circ)} \right) = 9
   \]
   Now, solve for \( h \):
   \[
   h = \frac{9}{\frac{1}{\tan(27^\circ)} - \frac{1}{\tan(53^\circ)}}
   \]

Let's compute this.The elevation of the plane is approximately \( 7.44 \) miles.

Would you like more details on the solution or have any further questions? Here are five related questions you might find interesting:

1. How does the tangent function relate to angles of depression and elevation?
2. What would change if the distance between the mileposts were different?
3. Can this method be applied if the angles of depression are given in radians?
4. How can the height be calculated if the pilot only knows the angle to one milepost?
5. How can you generalize this approach to find distances in different trigonometric problems?

**Tip:** Always ensure your calculator is in the correct mode (degrees or radians) based on the problem's requirements when using trigonometric functions.

A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 9 miles apart, to be ∠x = 27° and ∠y = 53°. Find the elevation of the plane.

Solve a trigonometric problem involving angles of depression. A pilot observes two mileposts, 9 miles apart, with angles of 27° and 53°. Learn how to use the tangent function to find the plane's elevation. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation