Answer
Please see the work below.
Work Step by Step
(a) The impedance of the plant can be determined as follows:
$Z=\sqrt{R^2+X_L^2}$
We plug in the known values to obtain:
$Z=\sqrt{(25)^2+(45)^2}$
$Z=51.48\Omega$
(b) We know that
$cos\phi=\frac{R}{Z}$
$cos\phi=\frac{25}{51.48}$
$cos\phi=0.486$
(c) The rms current through the plant can be determined as
$I_{rms}=\frac{V_{rms}}{Z}$
We plug in the known values to obtain:
$I_{rms}=\frac{485}{51.48}$
$I_{rms}=9.42A$
(d) We know that
$R=\sqrt{R^2+(X_L-X_C)^2}$
This simplifies to:
$X_L=X_C$
$\implies 45=\frac{1}{120\pi c}$
$\implies c=\frac{1}{45\times 120\pi}$
$\implies c=58.9\mu F$
(e) We know that
$I_{rms}=\frac{P_{avg}}{V_{rms}cos\phi}$
$\implies I_{rms}=\frac{2220}{485\times 1}$
$\implies I_{rms}=4.58A$
By comparing the currents found in part (c) and part (e), we conclude that $9.42A\gt 4.58A$.
