A flute is designed so that it plays a frequency of $261\mathrm{.}6$ Hz$,$ middle C$,$ when all the holes are covered and the temperature is $20\mathrm{.}0$C$.$

$($a$)$ Consider the flute to be a pipe open at both ends and find its length, assuming the middle$-$C frequency is the fundamental frequency. $($Give your answer to at least three decimal places.$)$

m

$($b$)$ A second player, nearby in a colder room, also attempts to play middle C on an identical flute. A beat frequency of $2\mathrm{.}83$ beats$/$s is heard. What is the temperature of the room?

$$C