Another problem on Thermal Expansion

[andrew410]

A beaker made of ordinary glass contains a lead sphere of diameter 4 cm firmly attached to its bottom. At a uniform temperature of -10 degrees Celsius, the beaker is filled to the brim with 118 cm^3 of mercury, which completely covers the sphere. How much mercury overflows from the beaker if the temperature is raised to 30 degrees Celsius?

Things I know so far:
average linear expansion coefficient of glass: 9*10^-6
average linear expansion coefficient of lead: 29*10^-6
average volume expansion coefficient of mercury: 1.82*10^-4
$$\Delta V=\beta V_{i}\Delta T$$

Wouldn't nothing spill because when the beaker's temperature rises then the beaker expands more.
If I'm wrong then could someone lead me into the right direction?
Thanks in advance! :)

 

[Doc Al]

Wouldn't nothing spill because when the beaker's temperature rises then the beaker expands more.

Don't guess, figure it out! How much does the beaker expand? The lead sphere? So... how much volume is left available in the beaker? How much does the mercury expand? So... how much must overflow?

Hint: Given the linear expansion coefficient, how can you find the volume expansion coefficient?

 

[andrew410]

How would you get the height of the beaker? I assume that the diameter of the beaker is approximately 4 cm because the sphere is firmly attached to the bottom of the beaker. Or is this the wrong way to get the initial volume of the beaker?

 

[Doc Al]

You have the initial volume of mercury and the dimensions of the sphere. That's all you need to find the volume of the beaker.

 

[andrew410]

So the volume of the beaker is volume of mercury + volume of the sphere?

 

[Doc Al]

So the volume of the beaker is volume of mercury + volume of the sphere?

Right. At least initially.