Impedance, resistance, capacitive reactance, and inductive reactance are related by:

$\overline{){\mathbf{Z}}{\mathbf{=}}\sqrt{{\mathbf{R}}^{\mathbf{2}}\mathbf{+}{\mathbf{(}\mathbf{X}}^{}}}$_{L}_{C}

Inductive reactance:

$\overline{){{\mathbf{X}}}_{{\mathbf{L}}}{\mathbf{=}}{\mathbf{2}}{\mathbf{\pi}}{\mathbf{f}}{\mathbf{L}}}$, where f is the frequency and L is the inductance.

Capacitive reactance:

$\overline{){{\mathbf{X}}}_{{\mathbf{C}}}{\mathbf{=}}\frac{\mathbf{1}}{\mathbf{2}\mathbf{\pi}\mathbf{f}\mathbf{C}}}$, where C is the capacitance.

Power:

$\overline{){{\mathbf{P}}}_{\mathbf{a}\mathbf{v}\mathbf{g}}{\mathbf{=}}\frac{{{\mathbf{V}}_{\mathbf{r}\mathbf{m}\mathbf{s}}}^{\mathbf{2}}}{\mathbf{Z}}{\mathbf{c}}{\mathbf{o}}{\mathbf{s}}{\mathbf{\varphi}}}$

A RLC series circuit with an AC voltage source has a 200 Ω resistor and a 25.0 mH inductor. At 8000 Hz, the phase angle is 45.0°.

(a) What is the impedance?

(b) Find the circuit’s capacitance.

(c) If V_{rms} = 408 V is applied, what is the average power supplied?

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