Answer
$16^{\circ}$F
Work Step by Step
Since December $31$ represents Day $365$, we substitute $x=365$ in the formula:
$T(365)=37+21\sin[\frac{2\pi}{365}(x-91)]$
$T(365)=37+21\sin[\frac{2\pi}{365}(365-91)]$
$T(365)=37+21\sin[\frac{2\pi}{365}(274)]$
$T(365)=37+21\sin[4.717]$
$T(365)=37+21(-1.000)$
$T(365)=37-21$
$T(365)=16^{\circ}$F
Therefore, on Day $365$, the temperature is $16^{\circ}$F.