# In a certain city, the temperature (in F^{\circ}) at t hours aft

In a certain city, the temperature (in ${F}^{\circ }$) at t hours after 9 AM is modeled by the function $T\left(t\right)=50+14\mathrm{sin}\left(\frac{\pi t}{12}\right)$. Find the average temperature during the period from 9 AM to 9 PM.
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Step 1
The temperature function is given as
$T\left(t\right)=50+14\mathrm{sin}\left(\frac{\pi t}{12}\right){F}^{\circ }$
t is time in hours
The analysis is started at 9 AM so lete a that instant $t=0$
At 9 PM, $t=12$
Step 2
The average value of a function is given by
${f}_{av}=\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)dx$
${T}_{av}=\frac{1}{12-0}{\int }_{0}^{12}\left[50+14\mathrm{sin}\left(\frac{\pi t}{12}\right)\right]dt$
$=\frac{1}{12}{\left[50t+14\left(\frac{-\mathrm{cos}\frac{\pi t}{12}}{\frac{\pi }{12}}\right)\right]}_{0}^{12}$
$=\frac{1}{12}\left[15\left(12-0\right)-14\left(\frac{12}{\pi }\right)\left\{\left(\mathrm{cos}\frac{\pi ×12}{12}\right)-\mathrm{cos}0\right\}\right]$
$=15-\frac{14}{\pi }\left[\mathrm{cos}\pi -\mathrm{cos}0\right]$
$=15-\frac{28}{\pi }$
${T}_{av}={6.0873}^{\circ }F$