Question

Scientists want to place a 3100 kg satellite in orbit around Mars. They plan to have the satellite orbit a distance equal to 1.9 times the radius of Mars above the surface of the planet. Here is some information that will help solve this problem:
mmars = 6.4191 x 1023 kg
rmars = 3.397 x 106 m
G = 6.67428 x 10-11 N-m2/kg2
1)What should the radius of the orbit be (measured from the center of Mars), if we want the satellite to take 8 times longer to complete one full revolution of its orbit?
so far I have found speed to orbit 2085m/s and 8.2hrs to complete one revolution
I have no clue how to solve this or what to plug in where

Answer 1

Ac = omega^2 R
m Ac = G m Mmars /R^2
so
omega^2 R = G Mmars/R^2
omega^2 = G Mmars/R^3
a circle is when omega T = 2 pi
or
T ^2 = 4 pi^2/omega^2 = 4 pi^2 R^3 /G Mmars
Tbig^2 = 64 T^2
so
Rbig^3 = 64 R^3
Rbig/R = 64^(1/3) or 4 times the radius