Intro to arc length

Question

A weather satellite orbits the earth in a circular orbit 500 miles above the earth's surface. What is the radian measure of the angle (measured at the center of the earth) through which the satellite moves in traveling 680 miles along its orbit? (The radius of the earth is 3960 miles.)

Answer 1

The radius r of the circular arc is the radius of the earth plus the 500 miles above the earth. The distance d travelled along an arc of a circle of radius r is equal to the angle in radians times the radius, r*a, where a is the angle travelled in radians. Since d=ra then a = d/r. You will have to figure out what d and r are from the statement of the problem and then do the division.

Answer 2

We know that the full revolution of a circle in radians is 2*pi. First we have figure out what fraction of the circle the satellite has traveled so that we can convert it to radians. To do that we first need the circumference (C) of the orbit. If the satellite is 500 miles above Earth's surface, then radius of the satellite's orbit = Earth's radius + 500. They give us Earth's radius as 3960 miles. R_satellite = 500 + 3960 = 4460. Then we must calculate the circumference. C = 2*pi*r -> C = 2*pi*4460 -> C = 8920*pi

Now we look at the amount of travel (680 miles) vs the total circumference (8920*pi miles) -> 680/(8920*pi) = 17/(223*pi). To convert this to radians we must multiply by 2*pi (full revolution of a circle) -> 17/(223*pi) * 2*pi

34/233 radians.