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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

In a certain city the temperature ( in  F ) t hours after 9 AM was modeled by the function

T ( t ) = 50 + 14 sin π t 12

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

To determine

To find:

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

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-aabfx 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=112-0012Ttdt

=11201250+14sinπt12dt

=11201250 dt+01214sinπt12dt

Integrating,

=11250t012

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