Thermal Expansion of Brass piston in Steel cylinder

[NickP89]

Homework Statement

You have been assigned to design brass pistons to slide inside steel cylinders. The engines in which these pistons will be used will operate between 20.0 degrees Celsius and 150.0 degrees Celsius. Assume that the coefficients of expansion are constant over this temperature range.
If the cylindrical pistons are 25.000 cm in diameter at 20.0 degrees C, what should be the minimum diameter of the cylinders at that temperature so the pistons will operate at 150.0 degree C?

Homework Equations

ΔV=βV(Initial)ΔT for Volume Expansion, where β is the coefficient of volume expansion
ΔA=2αA(initial)ΔT for Area Expansion, where α is the coefficient of linear expansion

The coefficient of volume expansion for brass is 6*10^-5 and for steel is 3.6*10^-5. The coefficient of linear expansion for brass is 2*10^-5 and for steel is 1.2*10^-5.

Volume of cylinder: pi*r^2*h

The Attempt at a Solution

I am not sure if I have to use the volume expansion equation or the area expansion equation. I understand that the cylinders are solids and would undergo volume expansion but I wasn't given a height of the cylinder. I tried assigning a value of 1cm for the height but I don't think that's O.K. I tried using the area expansion but also got numbers that were not realistic.

Attempt at solution:

ΔV=(6*10^-5)(130K)(pi*(12.5cm)^2*h)=(3.83cm)h for the brass piston

The steel cylinder would also expand:

ΔV=(3.6*10^-5)(130K)(pi*r^2*h)=0.0147*r^2*h

I really don't know where to go from here, I know that the brass will expand more than the steel, and that the steel cylinder certainly has to be bigger than the brass piston at 20 degrees celsius, but I don't know how to relate the two. Any direction and help will be appreciated very much.

 

[nure]

Isn't the case that the when the brass piston reaches 150 degree celsius, then the inner diameter of the steel cylinder(also at 150 deg) must be the same as the outer diameter of the piston? There will be no gap between the piston and the cylinder at 150 deg, but there will be at 20 deg. You must design the cylinder so that the brass piston is allowed to expand freely?