RLC Circuit- Find average power in circuit

[yankeekd25]

Homework Statement

When the power factor of RLC circuit is equal to one, the frequency of the voltage source is 3 x 10^3 Hz. The rms value of the voltage source is 139 Volts and at a frequency of 3 x 10^3 Hz, the rms current in the circuit is 37.1 amps. If the inductive reactance at 3 x 10^3 Hz is 47 Ohms, what is the average power of the circuit in Watts at 0.78 times the resonant frequency of the circuit?

Homework Equations

Pav = Vrms Irms cos (|)
Vrms= V/sqrt 2
Irms= I/sqrt 2
V= XL I
V= Xc I
XL= 2 pi f L
Xc= 1/ 2 pi C
Res freq= 1/2pi 1/ sqrt (LC)

The Attempt at a Solution

I'm going in circles with this problem. Given the RMS value of the voltage and current, I tried to find the max voltage and current, and then plug that into V=XL I and V= Xc I to find XL and Xc, but I am probably wrong.

I guess I need to use the XL and Xc for each at the given frequency, to find both L and C to find the resonance frequency, multiply that number by .78, and then plug everything back into something?

Can someone please help me sort out this problem? Thanks

 

[yankeekd25]

Can someone please help me out and get me started in the right direction please?

 

[GRB 080319B]

If the power factor cos($$\varphi$$) = 1, then the phase angle between the Vmax and Imax vectors in the phasor diagram must be $$\varphi$$ = 0. This means that Vmax vector has no y component, which means XL = XC. Since we know the inductive reactance and the resonant frequency, we can find the inductance and capacitance of the circuit. Also, at the resonant frequency, the impedance z = sqrt( R^2 + (XL -XC)^2 ) = sqrt( R^2 ) = R, so you can use Vrms and Irms to find R ( Vrms = Irms*z = Irms*R ). Then, you can find the reactances and impedance at the new frequency, draw the phasor/impedance diagram, find the new phase angle $$\varphi$$ between V and I ( $$\varphi$$ = tan^(-1)( (XL -XC) / R), and use those to calculate Pav.