Different Voltage for the Same Power

The key to solving this one is to think of the sinusoids as complex numbers. Note that in the first case, $v_S$ and $v_R$ have the same magnitude, and $v_S$ lags $v_R$ by 90 degrees. Find the voltage difference in case 1:

Case 1:

$\Delta \, v = v_S - v_R = 1 \angle {-90^\circ} - 1 \angle 0^\circ =\sqrt{2} \angle {-135^\circ}$

In case 2, the two sinusoids have the same phase, so the voltage difference is more straightforward.

Case 2:

$\Delta \, v = v_S - v_R = \alpha \angle {0^\circ} - 1 \angle 0^\circ = (\alpha - 1) \angle 0^\circ$

In order for the same average power to be dissipated in both cases, the magnitudes of the voltage differences must be the same. Also, we know that $\alpha$ is a positive number:

$\alpha - 1 = \sqrt{2} \\ \alpha = 1 + \sqrt{2} \approx 2.4142$