Satellite
Rationale
This task allows students to use trigonometry and properties of circles and triangles to find distances.
Instructional Task
A satellite orbiting the earth uses radar to communicate with two control stations on the earth’s surface. The satellite’s orbit maintains a 10-degree angle of separation between the two stations. Knowing that the earth’s radius is 3,963 miles, answer the following questions. Round all answers to the nearest whole number.
- How many miles will a signal sent from Station 1 to the satellite and then to Station 2 have to travel? Explain your answer.
- A satellite technician is traveling from one station to the other in a direct path along the earth’s surface. What is the distance she will have to travel? If she travels an average of 50 mph, how long will the trip take? Explain your answer.
- If a signal could travel through the earth’s surface from one station to the other, what is the shortest distance the signal could travel to get from Station 1 to Station 2? Explain your answer.
Discussion/Further Questions/Extensions
Students might be asked to solve this problem in metric units (the radius of the earth is approximately 6378 km). Students could also use the law of sines or the law of cosines to answer question 3.
Sample Solutions
- How many miles will a signal sent from Station 1 to the satellite and then to Station 2 have to travel? Explain your answer.
The line from the satellite to Station 1 is tangent to the Earth, therefore it forms a right angle with the radius of the earth at that point. Because the triangle formed by the satellite, Station 1, and the center of the earth is a right triangle, then trigonometric ratios can be used to solve the problem. Let x equal the length from Station 1 to the satellite.
x = 22,475 milesUsing trigonometric ratios and the right triangle formed by the satellite, Station 1, and the center of the earth, the length of the hypotenuse (the center of the earth to the satellite) can be found. Since Station 2 is on the hypotenuse and on the earth’s surface, its distance from the satellite can be found by subtracting the radius from the length of the hypotenuse. Let y equal the length from the center of the earth to the satellite.
y = 22,822Distance from satellite to Station 2 = 22,822 – 3963 = 18,859 miles
The distance from Station 1 to the satellite is 22,475 miles and the distance from the satellite to Station 2 is 18,859 miles. Therefore the total distance from Station 1 to the satellite to Station 2 is 41,334.
- A satellite technician is traveling from one station to the other in a direct path along the earth’s surface. What is the distance she will have to travel? If she travels an average of 50 mph, how long will the trip take? Explain your answer.
Measure of central angle = 180 – (90 + 10) = 80°
The distance from Station 1 to Station 2 is 5,533 miles.
If the technician travels an average of 50 mph, it will take approximately 111 hours to get from Station 1 to Station 2.
- If a signal could travel through the earth’s surface from one station to the other, what is the shortest distance the signal could travel to get from Station 1 to Station 2? Explain your answer.
If the signal could travel through the earth’s surface, the distance the signal travels would be approximately 5095 miles. To find this distance, use the isosceles triangle with vertices at the center of the earth (point C), Station 1, and Station 2. The angle at the center of the earth is 80° and the other two angles each measure 50°. Construct a perpendicular bisector from point C to the line connecting Station 1 and Station 2. This creates two right triangles. Calculating
, you find the measure from the perpendicular bisector to Station 1 to be 2547.37. The distance between Station 1 and Station 2 is twice this distance, or approximately 5095 miles.