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Page 2
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 ℉.
James Z. In a certain city the temperature (in °F) t hours after 9 AM was modeled by the function . Find the average temperature Tave during the period from 9 AM to 9 PM. (Round your answer to the nearest whole number.)T(t) = 46 + 17 sin(πt/12)