$($b$)$ In the circuit shown in Figure $3\mathrm{.}2,$ the voltage source, $\backslash (\backslash$widehat$\{$v$\_\{$i n$\}\}\backslash )$ was measured using an oscilloscope. It was found to show a sinusoidal behaviour, with a frequency of $20$ kHz and a peak$-$to$-$peak voltage of $28\mathrm{.}2$ V $.$

rigure $\backslash (3.\backslash$angle $\backslash )$

i$)$ State the value of the load resistance, $\backslash (\backslash$boldsymbol$\{$R$\}\_\{$L$\}\backslash ),$ that would be required to achieve maximum power transfer into this load.

$[2$ marks$]$

ii$)$ Calculate the maximum power that can be transferred into this load resistor.

$[6$ marks$]$

iii$)$ One issue with achieving maximum power transfer is that the efficiency is not optimum. A particular design requires an efficiency of the power dissipated in the load to be at least $\backslash (80\backslash \%\backslash )$ of the total power supplied from the source. Calculate a load resistance value to achieve this.