Problem 30 - Precalculus: A Concise Course

Question

A Global Positioning System satellite orbits 12,500 miles above Earth's surface (see figure). Find the angle of depression from the satellite to the horizon. Assume the radius of Earth is 4000 miles.

Step-by-Step Solution

Answer

The angle of depression from the satellite to the horizon is approximately \(\text{tan}^{-1}\left(\frac{12500}{4000}\right)\) degrees.

Step 1: Draw a diagram

Start by drawing a diagram to illustrate the problem. Label the center of the Earth 'O', the satellite 'S', and the point where the line of sight to the horizon meets the Earth, 'H'. Now you have a right-angled triangle OSH.

Step 2: Formulate the equation

Next, we can label the sides of our triangle. The distance from the satellite to the Earth's surface is the line OS, which is the sum of the radius of the Earth, OH: 4000 miles, and the height of the satellite, SH: 12500 miles. Therefore, OS = 4000 + 12500 = 16500 miles. The line OH is the radius of the Earth, which is 4000 miles. Now, since the triangle is a right-angled triangle, the tangent of the angle SOH can be found using the formula: \(\text{tan}(\theta) = \frac{\text{Opposite side}}{\text{Adjacent side}}\). In this case, the opposite side is SH = 12500 miles and the adjacent side is OH = 4000 miles. Therefore, the formula becomes \(\text{tan}(\theta) = \frac{12500}{4000}\)

Step 3: Solve for the angle of depression

To find the angle, we take the inverse tangent or arctangent (often written as tan^-1 or atan) of both sides: \(\theta = \text{tan}^{-1}\left(\frac{12500}{4000}\right)\). Use a calculator to compute the value of \(\theta\). Ensure that your calculator is set to output in degrees, not radians.