Understanding the Concept:
The average electrical power ($P_{\text{avg}}$) consumed in an alternating current circuit depends on the root-mean-square (rms) voltage, rms current, and the phase difference ($\phi$) between them:
\[
P_{\text{avg}} = V_{\text{rms}} I_{\text{rms}} \cos\phi = \frac{V_0}{\sqrt{2}} \frac{I_0}{\sqrt{2}} \cos\phi = \frac{V_0 I_0}{2} \cos\phi
\]
Step 1: Identify peak values and find the relative phase angle difference ($\phi$).
From the given equations:
Peak Voltage, $V_0 = 0.5\text{ V}$
Peak Current, $I_0 = 0.5\text{ A}$
Phase configuration matching, $\phi_V = 80\pi t + \pi$ and $\phi_I = 80\pi t$
The net phase difference $\phi$ is:
\[
\phi = \phi_V - \phi_I = (80\pi t + \pi) - 80\pi t = \pi \quad (180^\circ)
\]
Step 2: Substitute values into the average power formula.
\[
P_{\text{avg}} = \frac{0.5 \times 0.5}{2} \times \cos(\pi)
\]
We know that $\cos(\pi) = -1$:
\[
P_{\text{avg}} = \frac{0.25}{2} \times (-1) = -0.125\text{ W}
\]
(Note: A negative power value indicates that the phase-inverted source configuration is structurally returning power back to the primary line environment network).