People,
I CAN'T believe you're STILL having problems with this material.
OK...I'll go over the solution, but we can't waste all this time
revisiting these sorts of simple problems or we'll never cover
any of the new material.
Alright. So, we are starting with 8 people, assuming spouses and
offspring carry a uniform coeficient of zero, thus allowing those
terms to be dropped from the equation.
Let Jim be denoted as J, and Keith, Kendy, Susan, Mary, Gwen,
Frank, and Stacey be denoted K_m, K_b, S_p, M, G, F, and S_f,
respectively.
We will set J = 1, to denote the Jim's room. As previously stated,
Jim's room, R[J], has 1 queen size bed, and we can set the occupancy
equal to 1.0 (I'm normalizing all of the values so they can be
represented as a real number from 0 to 1 inclusive). Frank's room
starts out in the initial state as being in the same equivalence class
as Jim's, since they both have single queen-size beds, in other words,
R[J], R[F] are elements of [q]. (Frank's occupancy coefficient is
NOT equivalent to Jim's. The proof of this is left to the reader.
Hint: the zero spouse weighting coefficient is no longer a valid
assumption.)
Now, we get to the heart of the matter, the equation for the suite,
which is in the equivalence class [s] (not to be confused with S_p
or S_f). First, we can simplify things by setting K_m equal to K_b
sine Keith and Kendy will be staying together in the same room, and
solve the two equations simultaneously, which we will denote by K.
Thus R[K_m] = R[K_b] or 1 = R[K_b]/R[K_m]. And then we can simplify
it as 0 = (R[K_b]/R[K_m]) - 1 (we will refer to this as equation 1).
Now, recall that while Susan and Stacy do NOT yet have assigned rooms,
they CAN only be in one room at a given time, hence their INDIVIDUAL
occupency coefficents are both 1.0 (now it should be clear why I chose
to normalize all the results of the earlier steps).
So, you know the occupancy coefficents of Jim, Susan, Stacey. Keith and
Kendy's can be easily derived and Frank's can be set as a dependent
variable to one of the earlier equations. From that, you can derive
a weighted, normalized value for the number of people that are there,
and then, as a final step, you can plug in the actual spouse/offspring
coeffients and calculate the resultant number of rooms, room assignements,
as well as home many people are actually in the suite.
Now I've spent too much time on this material already, it's time to move
on. Oh yes, the bonus part: remember, while, in theory, you can keep
adding cots to a room, there IS an upper bound to it, which the lesser
of the number of people or the number of cots. Now if you use L'Hopital's
rule and Bernoulli's Principal, you should be able to get the answer.
Oh, and you might as well skip the part involving complex numbers and
just stick with real numbers.
--Doctor Asshole, PhD.