Calculating Orbital Radius of Ganymede using Orbital Period

[trekkie1701c]

Homework Statement

"If Ganymede, one of Jupiter's moons, has a pariod of 7.15 days, how many uits are there in it's orbital radius? Use the information given in Example Problem 1."

Example Problem:

"Galileo measured the orbital sizes of Jupiter's moons using the diameter of Jupiter as a unit of measure. He found that Io, the closest moon to Jupiter, had a period of 1.8 days and was 4.2 units from the center of Jupiter. Callisto, the fourth moon from Jupiter, had a period of 16.7 days. Using the same units that Galileo used, predict Callisto's distance from Jupiter."

Homework Equations

(Ta/Tb)^2=(Ra/Rb)^3

The Attempt at a Solution

My solution to the example problem:

Ta = 1.8
Tb = 16.7
Ra = 4.2

(1.8/16.7)^2 = (4.2/Rb)^3

.0116 = 74.088(Rb^3) (Dividing 1.8 by 16.2 and squaring, and cubing 4.2 and Rb)
.2264Rb=4.2 (Multiplying by Rb^3 and then taking the cubed root of everything)
Rb=18.5512 (Dividing by .2264)

I considered that since the book gave an answer of 19, that I was close enough, considering the book's want to round everything.

For the problem I'm having difficulty with, I did this:

Ta = 1.8
Tb = 7.15
Ra = 4.2

(1.8/7.15)^2 = (4.2/Rb)^3
.2517^2=4.2^3/Rb^3 (1.8 divided by 7.15)
.0634=4.2^3/Rb^3 (Squaring previous answer)
.0634(Rb^3)=4.2^3 (Multiplying by Rb^3)
.3987Rb=4.2 (Taking the cubed root of both sides)
Rb=10.5342 (Dividing by .3987)

The book, however, lists an answer of exactly 4. I can't figure out where I messed up.

 

[link2001]

Well, if Tb > Ta then Rb > Ra. In other words, the text must have the wrong answer. (My result is the same as yours).

Next time, try to do all of the algebric manipulations before punching the numbers into the calculator. There are a couple of good reasons for doing this: you are less likely to make an error with the calculator, mistakes are a lot easier to track down and it makes it easier for the marker/fellow students to follow your steps (symbols are easier to follow than numbers).

Hope it helps.

 

[D H]

Are you sure you are reading the problem correctly? A body with an orbital radius of 4 units would obviously have a shorter period than a body with a radius of 4.2 units.

 

[trekkie1701c]

I read it correctly. The problems I listed above are word for word what's in the textbook.

And, link2001. Thanks; I'll start doing that from now on.