Given a uniform resistive wire with resistance $12.$

$($a$)$ If the wire is bent to form an equilateral triangle, what is the equivalent resistance equilateral triangle $($each side has equal length$)$ across two of the corners?

$($b$)$ If the wire is bent to form a regular triangular pyramid, what is the equivalent resistance across two of the apexes?

regular triangular pyramid $($each side has equal lengh$)$

$($c$)$ The wire is melted and recast to a thicker uniform wire of half of the original length. A $3\mathrm{.}0-V$ battery is connected across the two ends of the new wire. What is the power output of the battery?

$[20$ marks$]$