#### Answer

$37^{\circ}$F

#### Work Step by Step

Since October $1$ represents Day $274$, we substitute $x=274$ in the formula:
$T(274)=37+21\sin[\frac{2\pi}{365}(x-91)]$
$T(274)=37+21\sin[\frac{2\pi}{365}(274-91)]$
$T(274)=37+21\sin[\frac{2\pi}{365}(183)]$
$T(274)=37+21\sin[3.150]$
$T(274)=37+21(-0.0084)$
$T(274)=37-0.177$
$T(274)=36.82\approx37^{\circ}$F
Therefore, on Day $274$, the temperature is $37^{\circ}$F.