"In January 2006, astronomers reported the discovery of a planet comparable in size to the Earth orbiting another star and having a mass of about 5.5 times the Earth's mass (5.97 x 10^24 kg). It is believed to consist of a mixture of rock and ice, similar to Neptune. If this planet has the same density as Neptune (1.76g/cm^3), what is the radius expressed a) in kilometers, b) in miles"
What I know is that the mass is ~5.97 x 10^24(5.5) = 3.2835 x 10^26 and I tried to convet to grams and got 3.2835 x 10^29 grams. Density is 1.76 g/cm^3[/B]
The question didn't provide equations so I resorted to google. I know that density = mass/volume and I found the equation for the volume of a sphere to be V = 4/3pir[/B]
The Attempt at a Solution
I started by multiplying the mass of the Earth by 5.5 and got 3.2835 x 10^26 kg, I then converted to grams for division to be compatible with the density of Neptune. So in grams I believe it is 3.2835 x 10^29.
Rearranging d=m/v I get v=m/d. I divide 3.2835 x 10^29 g by 1.76 g/cm^3 and get 1.865625 x 10^29 g/cm^3.
I then take this value for volume and plug it into V=4/3pir.
1.865625 x 10^29 = 4/3pir (this is the same as V = 4pir/3 ??)
I multiply both sides by 3 to remove the denominator of 3 and get 5.596875 x 10^29 = 4pir
I divide both sides by 4pi to isolate r and I get 4.453851611 x 10^28 = r
I apply a cube root to the radius to get 1.645 x 10^29 cm
This is where I get confused. The radius I am getting is no where near the radius of Neptune, despite having the same density? 1.645 x 10^29 cm is an absurd number, so I consulted with wolfram alpha and got a more reasonable number of 3.544 x 10^9.
My calculator says that the cube root of 4.453 x 10^28 is 1.645 x 10^29. Wolfram alpha says that it is 3.544 x 10^9.
Just because the planet has the same density as Neptune, does that mean the radius should be in the same ballpark? My current answer for the radius in km is 35,440,000.
I think I am comfortable enough with algebra to more terms around in these equations to get what I want, but the numbers involved are screwing me up? It's very possible that I made mistakes converting as well. Not sure what the deal is with the differences in cube roots from my calculator vs wolfram alpha.
Any help is GREATLY appreciated.[/B]