internal resis. prob.?

Question

When a battery sends current through a resistace R1 for time t,the heat produced in the resistor is Q.When the same battery sends current through another resistance R2 for time t,the heat produced in R2 is again Q.The internal resistance of battery is-
1. (R1R2\2)^1\2
2. R1R2^1\2
3. R1-R2\2
4. R1+R2\2

Answer 1

I1 = V/(R1 + Ri)
Q = I1^2R1 = (V/(R1 + Ri)^2R1
I2 = V/(R2 + Ri)
Q = I2^2R2 = (V/(R2 + Ri))^2R2
R1(R2 + Ri)^2 = R2(R1 + Ri)^2
R1R2^2 + 2R1R2Ri + R1Ri^2 = R2R1^2 + 2R1R2Ri + R2Ri^2
R1R2^2 + R1Ri^2 = R2R1^2 + R2Ri^2
Ri^2(R1 - R2) = R1R2(R1 - R2)
Ri^2 = R1R2
Ri = √R1R2
which seems to agree with your answer #2.