A surveillance satellite circles the earth at a hight of h miles above the surface. Suppose that d is the distance, in miles, on the surface of the earth that can be observed from the satellite. find an equation that relates to central angel (theta) to the hight h.
To find an equation that relates the central angle (θ) to the height (h) of the surveillance satellite, we can use the concept of similar triangles.
Consider a right triangle with the satellite at the top vertex, the center of the Earth at the bottom vertex, and a point on the Earth's surface at the other vertex. Let's call the length of the adjacent side of this triangle (the distance from the center of the Earth to the point on its surface) as "r" (the radius of the Earth).
Now, let's focus on the smaller right triangle formed by the satellite, the center of the Earth, and the point on the Earth's surface. The angle at the center of the Earth is equal to θ since it is the central angle.
By the definition of trigonometry, the tangent of an angle is equal to the ratio of the length of the opposite side to the length of the adjacent side. In this case, the opposite side is the height of the satellite (h), and the adjacent side is the radius of the Earth (r).
Therefore, we can write the equation as:
tan(θ) = h / r
Simplifying this equation gives us the relation between the central angle (θ) and the height (h) of the surveillance satellite:
θ = arctan(h / r)
where arctan represents the inverse tangent function.
To find an equation that relates the central angle (theta) to the height (h) of a surveillance satellite, we need to consider the geometry of the situation.
Let's assume that the radius of the Earth is denoted by R (approximately 3959 miles). Since the satellite is h miles above the surface of the Earth, the distance from the satellite to the center of the Earth is R + h.
Now, let's consider a right triangle formed by the center of the Earth, the satellite, and a point on the Earth's surface directly below the satellite. The hypotenuse of this triangle is the distance from the satellite to the center of the Earth, which is R + h. The base of the triangle is the radius of the Earth, R.
According to geometry, the central angle (theta) can be defined as the angle between the two radii that form the triangle. This central angle corresponds to the angle formed between the base and the hypotenuse of the right triangle.
Using trigonometry, we can relate the central angle (theta) to the height (h) of the satellite by defining the sine of the central angle:
sin(theta) = opposite/hypotenuse
sin(theta) = R/(R + h)
This equation relates the central angle (theta) to the height (h) of the satellite, with the assumption that R is the radius of the Earth (approximately 3959 miles).
If d is the distance between the two points of line of sight tangency, then d = (2µ/360)(2Pir)
Pi = 3.14
r = the radius of the Earth = 3963 miles
µ = arccos(r/(r + h)