Final answer:
To find the average power delivered to the load, we need to find the rms current and phase angle for both the resistor and the capacitor. We can then use these values to calculate the power for each component and sum them up to find the average power delivered to the load.
Step-by-step explanation:
The average power delivered to the load can be found using the formula P = Vrms * Irms * cos(theta), where Vrms is the voltage rms, Irms is the current rms, and theta is the phase angle between the voltage and current. In this case, the load consists of a resistor and a capacitor in parallel. The current through the resistor and capacitor will have different phase angles. To find the power, we need to find the rms current for each component and the phase angle for each component. Finally, we can calculate the power by summing up the powers of the resistor and capacitor.
First, let's find the rms current for the resistor. Since the resistor and capacitor are connected in parallel, the voltage across both components will be the same. We can use the given voltage source vt = 30 cos 1000t to find the rms voltage. The rms voltage, Vrms, is equal to the amplitude of the voltage, which is 30 V, divided by sqrt(2). Therefore, Vrms = 30 / sqrt(2) V.
Next, we need to find the rms current for the resistor, Irms_r. The current through the resistor will be in phase with the voltage, so the phase angle, theta_r, will be 0 degrees. Therefore, the power for the resistor, Pr, can be calculated using the formula Pr = Vrms * Irms_r * cos(theta_r).
Now let's find the rms current and phase angle for the capacitor, Irms_c and theta_c, respectively. The current through the capacitor will lead the voltage by 90 degrees. Therefore, the phase angle for the capacitor, theta_c, will be -90 degrees. The rms current for the capacitor, Irms_c, can be calculated using the formula Irms_c = (Vrms / Xc), where Xc is the capacitive reactance of the capacitor. The capacitive reactance, Xc, can be calculated using the formula Xc = 1 / (2 * pi * f * C), where f is the frequency and C is the capacitance.
Once we have the rms currents and phase angles for the resistor and capacitor, we can calculate the powers, Pr and Pc, for the resistor and capacitor, respectively. Finally, we can calculate the average power delivered to the load by summing up the powers of the resistor and capacitor, Pavg = Pr + Pc.