The temperature in Fairbanks is approximated by \( T(x)=37 \sin \left[\frac{2 \pi}{365}(x-101)\right]+25 \) where \( T(x) \) is the temperature on day \( x \), with \( x=1 \) corresponding to Jan. 1 and \( x=365 \) corresponding to Dec. 31 . Estimate the temperature on day 323 .

To find the temperature on day 323, we can substitute \( x = 323 \) into the equation \( T(x) = 37 \sin \left[\frac{2 \pi}{365}(x-101)\right]+25 \). Calculating \( T(323) \): \[ T(323) = 37 \sin \left[\frac{2 \pi}{365}(323-101)\right] + 25 \] \[ = 37 \sin \left[\frac{2 \pi}{365} \times 222\right] + 25 \] Now, calculate the argument of the sine function: \[ \frac{2 \pi}{365} \times 222 \approx 3.825 \] Now find \( \sin(3.825) \): Using a calculator, \( \sin(3.825) \approx -0.554 \). Substituting that back into our expression for \( T(323) \): \[ T(323) \approx 37 \times (-0.554) + 25 \] \[ \approx -20.478 + 25 \] \[ \approx 4.522 \] So, the estimated temperature on day 323 is approximately \( 4.5^\circ \text{C} \).