Write the standard form of the
equation for the path of a communications satellite in a circular
orbit 22,000 miles above Earth. (Assume that the radius of
Earth is 4000 miles.)
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To find the standard form equation for the path of a communications satellite in a circular orbit 22,000 miles above Earth, we can start by understanding the equation for a circle. The equation for a circle centered at the origin (0,0) with a radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) represents the coordinates of the center of the circle.
In this case, the satellite is in a circular orbit 22,000 miles above Earth. Since the radius of Earth is given as 4000 miles, the radius of the circular orbit can be calculated by adding the altitude of the satellite (22,000 miles) to the radius of Earth (4000 miles):
Radius of the orbit = 22,000 miles + 4000 miles = 26,000 miles
Since the satellite is in a circular orbit, we can assume that the center of the circle lies at the center of the Earth, which is the origin (0,0).
Now we can substitute these values into the equation:
(x - 0)^2 + (y - 0)^2 = (26,000 miles)^2
Simplifying, we get:
x^2 + y^2 = 676,000,000 miles^2
So, the standard form equation for the path of the communications satellite in a circular orbit 22,000 miles above Earth is x^2 + y^2 = 676,000,000 miles^2.