Problem 65 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Distance between a Satellite and a Tracking Station \(\mathrm{A}\) satellite traveling in a circular orbit 1600 kilometers above Earth is due to pass directly over a tracking station at noon. Assume that the satellite takes 2 hours to make an orbit and that the radius of Earth is 6400 kilometers. Find the distance between the satellite and the tracking station at 12: 03 P.M. (Source: NASA.) (figure cannot copy)

Step-by-Step Solution

Answer

The distance between the satellite and the tracking station at 12:03 P.M. is calculated using the law of cosines with a specific angle.

Step 1: Understand the Satellite's Orbit

The satellite travels in a circular orbit 1600 km above Earth. The total radius from the Earth's center to the satellite is the Earth's radius plus the altitude of the satellite, which is 6400 km + 1600 km = 8000 km. The satellite completes one orbit in 2 hours.

Step 2: Determine the Angle for 3 Minutes

The complete orbit takes 2 hours, equivalent to 120 minutes. In 3 minutes, the satellite covers a fraction of the orbit: \( \frac{3}{120} = \frac{1}{40} \) of the orbit. This fraction of the orbit corresponds to the angle \( \theta \) in radians, where \( \theta = 2\pi \times \frac{1}{40} = \frac{\pi}{20} \).

Step 3: Calculate the Distance Using the Law of Cosines

To find the distance between the satellite and the tracking station on Earth's surface, apply the law of cosines. This geometric relation for a triangle with sides 6400 km (Earth radius), 8000 km (radius from Earth's center to satellite), and angle \( \theta = \frac{\pi}{20} \) between them is given by:\[d^2 = 6400^2 + 8000^2 - 2 \times 6400 \times 8000 \times \cos\left(\frac{\pi}{20}\right)\]Solving for \(d\):\[d = \sqrt{6400^2 + 8000^2 - 2 \times 6400 \times 8000 \times \cos\left(\frac{\pi}{20}\right)}\]

Step 4: Perform Calculations

Calculating the above equation:1. \(6400^2 = 40960000\)2. \(8000^2 = 64000000\)3. Compute \(2 \times 6400 \times 8000 = 102400000\)4. Calculate \(\cos(\frac{\pi}{20})\), typically using a calculator.5. Substitute these into the cosine equation to find \(d\):\[d = \sqrt{40960000 + 64000000 - 102400000 \times \cos\left(\frac{\pi}{20}\right)}\]Complete the calculations to find \(d\).

Step 5: Conclusion

After performing the calculations, you find the final distance \(d\) between the satellite and the tracking station after accounting for 3 minutes of satellite travel.

Key Concepts

circular orbit

A circular orbit is a path that takes the form of a perfect circle around a central point, such as a planet or star. In this problem, the satellite is orbiting Earth in a circular manner. The altitude of the orbit is 1600 kilometers above Earth's surface. This is added to Earth's radius of 6400 kilometers, resulting in a total orbital radius of 8000 kilometers.

To complete one full orbit, the satellite travels all 360 degrees around Earth, which correlates to an angle of \(2\pi\) radians. The satellite completes this circular journey in precisely 2 hours. With the consistency of speed in a circular orbit, calculating the satellite's position at any given time becomes straightforward, which is crucial for determining the exact location at 12:03 P.M.

law of cosines

The law of cosines is a powerful tool in trigonometry for solving distances in triangles. This formula is particularly helpful when you know two sides of a triangle and the angle between them, allowing you to calculate the third side. The formula is given by:

• \(c^2 = a^2 + b^2 - 2ab\cos(C)\)

Here, \(c\) is the side you want to find, and \(a\), \(b\), and \(C\) are the known sides and the angle between them, respectively.

In the satellite scenario, the triangle consists of:

• Earth's radius (6400 km),
• The radius from Earth's center to the satellite (8000 km),
• The angle \(\theta = \frac{\pi}{20}\) radians,

By plugging values into the law of cosines, you can calculate the distance \(d\) from the satellite to the tracking station as soon as you've determined the necessary cosine value. Performing these calculations gives a precise measure of the satellite's distance from the ground point.

angle in radians

An angle measured in radians is a fundamental concept in trigonometry, contrasting with degrees. Radians offer a more natural measure related to the radius of a circle. One complete turn around a circle is \(2\pi\) radians, equivalent to 360 degrees.

To find out how much of the orbit the satellite covers in a particular time, you need to calculate the angle in radians for that time interval. Since it covers a fraction of the orbit in relation to 2 hours or 120 minutes in full, calculating the number of radians covered in 3 minutes involves:

• \(\theta = 2\pi \times \frac{3}{120} = \frac{\pi}{20}\)

Therefore, the small section of the orbit during this brief time is represented by an angle of \(\frac{\pi}{20}\). This angle measurement is integral for utilizing the law of cosines to find the desired distance in this application. Understanding radians thus simplifies the analysis and calculations involving periodic circular paths, such as the orbits of satellites.