Help!!! Planetary motion

[dudgey]

Please help!
Planet X rotates in the same manner as the earth, around an axis through its north and south poles, and is perfectly spherical. An astronaut who weighs 946 N on the earth weighs 920 N at the north pole of Planet X and only 854 N at its equator. The distance from the north pole to the equator is 1.889×104 km , measured along the surface of Planet X.

How long is the day on Planet X? Take free fall acceleration on Earth to be 9.80 m/s^2

I'm totally stuck and I've being trying to do this for about an hour in many different ways. The equation T^2=(4*pi^2*r^3)/G*M doesn't seem to work. I don't know what I'm doing wrong, and I don't understand what to do with the two different weights at the pole and equator. :(

[CartoonKid]

With the distance given you can calculate the radius of planet X.
Calculate the mass of the astronaut on the earth. Using this mass, you will find the gravitational acceleration of planet X. Then apply the formula T+mg'=mrw^2. Work out the w and eventually you will find T

[dudgey]

That's a different approach than what I was trying. How would I find w though from that information? There's something I'm missing on this question. I can do all the others that go along with it, just not this one. so frustrating...

[CartoonKid]

sorry. I need to clarify something. the former T and the latter T in my post is different one. The latter T means the time. The former T is given by the question. So you just need to put in everything to get the w.