mronjo7n
2021-11-16
Answered

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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George Blue

Answered 2021-11-17
Author has **18** answers

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

${f}_{av}=\frac{1}{b-a}{\int}_{a}^{b}f\left(x\right)dx$

${T}_{av}=\frac{1}{12-0}{\int}_{0}^{12}[50+14\mathrm{sin}\left(\frac{\pi t}{12}\right)]dt$

$=\frac{1}{12}{[50t+14\left(\frac{-\mathrm{cos}\frac{\pi t}{12}}{\frac{\pi}{12}}\right)]}_{0}^{12}$

$=\frac{1}{12}[15(12-0)-14\left(\frac{12}{\pi}\right)\{\left(\mathrm{cos}\frac{\pi \times 12}{12}\right)-\mathrm{cos}0\}]$

$=15-\frac{14}{\pi}[\mathrm{cos}\pi -\mathrm{cos}0]$

$=15-\frac{28}{\pi}$

${T}_{av}={6.0873}^{\circ}F$

The temperature function is given as

t is time in hours

The analysis is started at 9 AM so lete a that instant

At 9 PM,

Step 2

The average value of a function is given by

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