Two engineers, a physicist, and an orbital mechanist can't figure this out

Well, if you assume that the volume after the air is pumped out is known, then B] is the right answer. Let the atmospheric pressure be [itex]P_a[/itex] and the final pressure and final volume be [itex]P_f[/itex] and [itex]V_f[/itex]. Then:

[tex]
(2.25)P_a = P_f~V_f
[/tex]

Hence,

[tex]
P_f = \frac{(2.25)P_a}{V_f} = \frac{2.25}{k}
[/tex]

Do note here, that 'k' is not the constant you mentioned earlier i.e. [itex]P_1 V_1 = P_2 V_2 = k[/itex], as you said that it isn't given in the problem.

Also, unless the units of 'k' are given, nothing can be said about how the equation stands dimensionally. IMHO, this question seems quite incomplete.

 

What you are essentially saying is that:

[tex]
P_a - V_f > 0
[/tex]

Look at the operation on the left hand side. This operation is not dimensionally correct. You cannot subtract volume from Pressure.

And yes, the problem does suck since you have no idea what 'k' is given. If 'k' is said to be independent of the final volume, then my answer is wrong. Here, again, I do consider that the initial pressure is known. In our case, we have two unknowns [final pressure and final volume], but one equation only. Which leaves with either finding another equation, or knowing one of the variables, which is why I assumed that the final volume is known.

Again take the case as, we take the equation to be:

[tex]
\frac{2.25}{k}
[/tex]

also,

[tex]
\frac{p}{2.25}
[/tex]

is a solution, where, [itex]p = 6.025 / k[/itex]. Now, both of them are constants, and even the second solution is right. In this way you could come up with infinite solutions involving an arbitrary constant and the value '2.25'. Which is why this question is an epic fail.