Thermal Stress Problem- not sure how to attack this one

[RoKr93]

Homework Statement

The equation F/A = -Y\alpha\Delta T gives the stress required to keep the length of a rod constant as its temperature changes. Show that if the length is permitted to change by an amount \Delta L when its temperature changes by \Delta T, the stress is equal to F/A = Y(\Delta L/L_0 - \alpha\Delta T).

Homework Equations

thermal stress:
F/A = -Y\alpha\Delta T

solution:
F/A = Y(\Delta L/L_0 - \alpha\Delta T)

change in length:
\Delta L = \alpha L_0\Delta T

The Attempt at a Solution

I figure this will have something to do with the area, as that's the only relation to the length I can see in the problem. I tried substituting A0 + deltaA in for A in the thermal stress equation, but that didn't seem to get me anywhere. Would really appreciate some direction with this one.

 

[haruspex]

This is one of those questions where the answer seems so obvious that you are left wondering what method you are allowed to use. E.g., can you not argue that it's the same as allowing the rod to expand freely, then compressing it to length only ΔL longer than it started?