# A Problem from my book -9!

A conductor whose resistance is constant $$R= 5 \ \Omega$$ due to some reason is connected to a $$5 \ V$$ battery. The heat lost $$Q$$ to the surrounding due to radiations is dependent on both temperature of conductor $$T$$ at any instant and time elapsed $$t$$ as $$Q= a(T - T_0) +bt^2$$, where $$T_0$$ is temperature of conductor at time $$t=0$$, $$a= 10 \ J/K$$ and $$b= 4 \ J/s^2$$. The conductor has a constant heat capacity.

Find the time after which the temperature of conductor becomes $$T_0$$ again. If $$t= \dfrac XY$$, where $$X$$ and $$Y$$ are coprime integers, find $$X+Y$$.

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