$($a$)$ Use Figs. $31\mathrm{.}13$b and $31\mathrm{.}13$c to show that for a series

L$-$R$-$C circuit, V cosf $=$ IR$.$ Use this result in Eq$.(31\mathrm{.}31)$ to show that

the average power delivered by the source is Pav $=1$

$2$ I$2$R$.$ Therefore the

average power delivered by the source is equal to the average power con$-$

sumed by the resistor. This means the average power for the capacitor

and for the inductor is zero. These circuit elements consume electrical

energy during the part of the current oscillations when they are storing

energy, but all the stored energy in each is then released during another

part of the current oscillations. $($b$)$ In an L$-$R$-$C series circuit the ampli$-$

tude of the source voltage is $120$ V$,$ the source voltage leads the current

by $53\mathrm{.}1,$ and the average power supplied by the source is $80\mathrm{.}0$ W$.$ What

is the resistance R of the resistor in the circuit