PROFESSOR: Titan is the largest moon of Saturn. The mass of Titan is approximately 1.35 times 10 to the 23rd kilograms. The mass of Saturn is approximately 5.68 times 10 to the 26 kilograms. Titan takes approximately 16 days to orbit Saturn. Use this information, along with the universal gravitational constant g equals 6.67 times 10 to the negative 11 meters cubed per kilogram times seconds squared to estimate the distance from Titan to Saturn.

Now the equation we are going to use to calculate this is p squared equals numerator 4pi squared a cubed denominator g parentheses lowercase m plus capital M, where lowercase m is the mass of the smaller m and capital M is the mass of the larger body. And let's see, a is our-- a is the-- Oh, we're actually finding a. That's why I don't know what it is. OK.

Moving on from there, we know the orbit time. That's p. OK, there we go. So the orbit time of 16 days-- now we actually need to convert this into seconds because the gravitational constant is in seconds.

So that'd be 16 days multiply by 24 hours per one day multiplied by 3,600 seconds per 1 hour. And that will be equal to 1, 382,400 seconds. That's how long it takes to orbit Saturn.

Now let's fill in the information we know. So p squared, that is the-- that is our orbit time. So 1,382,400 squared equals numerator 4pi squared a cubed over denominator of g-- that's 6.67 times 10 to the negative 11 parentheses lowercase m plus capital M. So that is 1.35 times 10 to the 23 plus 5.68 times 10 to the 26 close parentheses.

So in order to find the distance from Titan to Saturn, we need to solve this equation for a. So I'm going to multiply by my denominator on both sides. And ultimately, we're going to take the cubed root of both sides as well. So a is equal to the cube root of a large fraction. Numerator is 1,382,400 squared parentheses 6.67 times 10 to the negative 11 close parentheses open parentheses 1.35 times 10 to the 23rd plus 5.68 times 10 to the 26 in the numerator.

And our denominator is 4pi squared. So a is approximately 1.224 times 10 to the 9, which is again, approximately 1,224,000 kilometers. So the distance between Titan and Saturn is approximately 1,224,000 kilometers.