#### To determine

**To find:**

The average temperature during the period from 9 AM to 9 PM.

#### Answer

≈59 ℉

#### Explanation

**1) Concept:**

Use the mean value theorem for integral to find the average temperature during the period from 9 AM to 9 PM.

**2) Mean Value Theorem for Integrals**

If f is continuous on a,b then there exists a number c in a,b such that

fc=fave=1b-a∫abfx dx

that is,

∫abfx dx=fcb-a

**3) Given:**

Tt=50+14sinπt12

**4) Calculation:**

During the period from 9 AM to 9 PM, it completes t=12 hours

The average temperature during the period from 9 AM to 9 PM means t=0 to t=12 is,

Ttave=112-0∫012Ttdt

=112∫01250+14sinπt12dt

=112∫01250 dt+∫01214sinπt12dt

Integrating,

=11250t012-14cosπt12π12012

=1125012-500-14cosπ1212π12-14cos0π12

=1125012-14cosππ12-14cos0π12

=1125012--14-14π12

=1125012--28π12

=112600+336π

=112600+106.95

=706.9512

≈59 ℉

**Conclusion:**

The average temperature during the period from 9 AM to 9 PM≈59 ℉.