Observers at locations due north and due south of a rocket launchpad sight a rocket at a height of 10 kilometers. Assume that the curvature of earth is negligible and that the rocket's trajectory at that time is perpendicular to the gound. How far apart are the two observers if their angles of elevation to the rocket are `80.5^(@) and 68.0^(@)`?

Options

A. 0.85 km
B. 4.27 km
C. 5.71 km
D. 20.92 km

Explanation

To solve the problem, we need to determine the distance between the two observers based on their angles of elevation to the rocket. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have two observers located due north and due south of a rocket launchpad. The rocket is at a height of 10 kilometers, and the angles of elevation from the observers to the rocket are 80.5 degrees and 68 degrees.

### Step 2: Draw a Diagram
Let's denote:
- Point C as the position of the rocket.
- Point A as the position of the observer to the north.
- Point B as the position of the observer to the south.

The height of the rocket (point C) is 10 km. The angles of elevation from points A and B to point C are 80.5 degrees and 68 degrees, respectively.

### Step 3: Set Up the Right Triangles
From the observers' points (A and B), we can form two right triangles:
- Triangle AOC (for observer A):
- Opposite side (height of the rocket) = 10 km
- Angle of elevation = 80.5 degrees
- Triangle BOC (for observer B):
- Opposite side (height of the rocket) = 10 km
- Angle of elevation = 68 degrees

### Step 4: Use Trigonometric Ratios
Using the tangent function, we can express the distances from the observers to the base of the rocket (points A and B) in terms of the angles of elevation:

1. For observer A:
\[ \tan(80.5^\circ) = \frac{10}{AC} \]
Rearranging gives:
\[ AC = \frac{10}{\tan(80.5^\circ)} \]

2. For observer B:
\[ \tan(68^\circ) = \frac{10}{BC} \]
Rearranging gives:
\[ BC = \frac{10}{\tan(68^\circ)} \]

### Step 5: Calculate Distances AC and BC
Now, we can calculate the distances using a calculator:

1. Calculate \( AC \):
\[ AC = \frac{10}{\tan(80.5^\circ)} \approx \frac{10}{5.144} \approx 1.94 \text{ km} \]

2. Calculate \( BC \):
\[ BC = \frac{10}{\tan(68^\circ)} \approx \frac{10}{2.475} \approx 4.04 \text{ km} \]

### Step 6: Find the Total Distance Between Observers A and B
The total distance \( AB \) between the two observers is the sum of distances \( AC \) and \( BC \):
\[ AB = AC + BC \approx 1.94 + 4.04 \approx 5.98 \text{ km} \]

### Conclusion
The distance between the two observers is approximately **5.98 kilometers**.