Thermodynamics Volume Expansion

[Ling_Ling]

An ordinary glass is filled to the brim with 360.0 mL of water at 100°C. If the temperature decreased to 18°C, how much water could be added to the glass?

Volume Expansion Coefficient for Water ß (C°)^-1 (I believe then, that I use Celsius and not Kelvin?)
ß(water) = 210E-6
ß(glass) = 9E-6
Vo = 360 mL
⌂T = -82°C (= -355°K) (or is it positive, since change is the absolute value of it?)

V = Vo(1+ß⌂T)

V = 360(1+210E-6*-82)
V = 360 - 360*210E-6*-82
V = 360 - 6.199
Interpreting this, I thought that there would be 6.2 mL more room for water (unless water increases in volume as it is cooled?)

For the container,
V = 360(1+9E-6*-82)
V = 360 - .2657
So .2657 mL more room in the container. Adding 6.2 + .2657 = 6.47 mL

I don't know whether this is right or wrong, and had a few similar attempts that were wrong, so am I correct in my thinking?

Also, I know that water is the only thing that contracts when heated and expands when cooled, but I didn't think this would make sense, since it would just be overflowing, when the question seems to imply that it is leaving room for more water.

[Doc Al]

So .2657 mL more room in the container. Adding 6.2 + .2657 = 6.47 mL

I don't know whether this is right or wrong, and had a few similar attempts that were wrong, so am I correct in my thinking?
For some reason you added the change in volume of the water and the glass. Both water and glass contract as they are cooled (at least in this range of temperatures), thus they both have less volume. If they both shrank by the same amount, then the glass would still be full. But the water shrinks more than the glass, leaving some extra space available. How much more?

[Ling_Ling]

Oh. I see now. So it would be:
6.2 mL - .2657 mL = 5.343 mL

I forgot that if the glass is contracting, there would be less room since it's the container of the liquid.

[Doc Al]

Oh. I see now. So it would be:
6.2 mL - .2657 mL = 5.343 mL
Right idea, but recheck your arithmetic.
I forgot that if the glass is contracting, there would be less room since it's the container of the liquid.
Right.