Solve the variation problem.
-The time in hours it takes a satellite to complete an orbit around the earth varies directly as the radius of the orbit (from the center of the earth)and inversely as the orbital velocity. If a satellite completes an orbit 680 miles above the earth in 9 hours at a velocity of 23,000 mph, how long would it take a satellite to complete an orbit if it is at 1700 miles above the earth at a velocity of 28,000 mph? Use 3960 miles as the radius of the earth, and round your answer to the nearest hundredth of an hour.
A)2.71 hours
B)9.02 hours
C)90.18 hours
D)18.48 hours

Question

Quizplus · Accepted Answer

The time $T$ it takes for a satellite to complete an orbit varies directly with the radius $r$ of the orbit and inversely with the orbital velocity $v$, which can be represented as $T = k \cdot \frac{r}{v}$, where $k$ is a constant of proportionality. Given the initial conditions, we can find $k$ and then use it to find the time for the new conditions. For the first satellite, $r_1 = 3960 + 680$ miles, $v_1 = 23,000$ mph, and $T_1 = 9$ hours. For the second satellite, $r_2 = 3960 + 1700$ miles, and $v_2 = 28,000$ mph. Solving for $k$ using the first set of conditions and applying it to the second set gives us the time for the second satellite, which rounds to 9.02 hours.

Solve the Variation Problem