Starting from a homework problem:
An aluminum cup of $100 cm^3$ capacity is completely filled with glycerin at $22°C$. How much glycerin, if any, will spill out of the cup if the temperature of both the cup and the glycerin is increased to $28°C$? (The coefficient of volume expansion of glycerin is $5.1x10^4/C°$.)
I find that I have the for efficient of linear expansion for aluminum, but I need to know how the volume of the cup changes. Worse, I don't know the dimensions of the cup.
I think I use the linear expansion equation for metal rod $\Delta L = L \alpha \Delta T$ to find how much taller the cup is after the temperature changed and the volume expansion equation for a solid of liquid $\Delta V = V \beta \Delta T$ but not knowing any of the dimension of the cup I do not see how to determine this?