the pilot in an airplane observes the angle of depression of a light directly below his line of sight to be 30.4 degrees. a minute later, its angle of depression is 43.0 degrees. if he is flying horizontally in a straight course at the rate of 150 mph, find the altitude at which he is flying?
To find the altitude at which the pilot is flying, we can use trigonometry and the concept of similar triangles.
Let's break down the problem step by step:
Step 1: Understand the problem.
We are given the angle of depression of a light observed by the pilot at two different times, along with the speed of the airplane. We need to find the altitude at which the pilot is flying.
Step 2: Draw a diagram.
Draw a diagram to visualize the situation. Label the necessary angles and distances. It will help you identify the relevant trigonometric relationships.
```
C
/|
/ |
h / |
/ | d1
/θ1_ |
A B
d2
```
In the diagram, the point A represents the position of the pilot, the point B represents the position of the light, and the point C represents the center of the Earth (assumed to be a flat surface for simplicity).
Step 3: Identify the given information.
We are given:
- Angle of depression at time 1, θ1 = 30.4 degrees
- Angle of depression at time 2, θ2 = 43.0 degrees
- Speed of the airplane, 150 mph
Step 4: Identify the variables to be determined.
We need to find:
- The altitude at which the pilot is flying, which is the distance h
Step 5: Set up the equations.
From the diagram, we can see that the triangle ABC is a right triangle, and we have two similar triangles.
From triangle ABC, we have:
tan θ1 = h / d1 (equation 1)
tan θ2 = h / d2 (equation 2)
Step 6: Solve the equations.
We have two equations with two unknowns (h, and d1 or d2). We can rearrange equation 1 to solve for d1 in terms of h:
d1 = h / tan θ1
Substituting this expression for d1 into equation 2, we get:
tan θ2 = h / (h / tan θ1)
tan θ2 = h * tan θ1 / h
tan θ2 = tan θ1
Since tan θ2 = tan 43.0 degrees and tan θ1 = tan 30.4 degrees, we can solve for h:
h = (d2 * tan θ1) / tan θ2
Step 7: Calculate the altitude.
Plug in the given values and calculate the altitude:
h = (150 mph * 1 hour) * tan 30.4 degrees / tan 43.0 degrees
Now, use the appropriate unit conversions to get the result in a desired unit (e.g., feet or meters).
Note: Make sure to convert the speed from mph to the appropriate unit (such as miles per minute) to match the time frame.
Calculating the exact value requires numerical calculations.