   Chapter 8.5, Problem 62E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Water Supply The flow rate R (in thousands of gallons per hour) of water at a pumping station during a day can be modeled by R = 53 + 7   sin ( π t 6 + 3.6 ) + 9   cos ( π t 12 + 8.9 ) , 0 ≤ t ≤ 24 where t is the time (in hours), with t = 0 corresponding to midnight.Find the average hourly flow rate from midnight to noon ( 0 ≤ t ≤ 12 ) .Find the average hourly flow rate from noon to midnight ( 12 ≤ t ≤ 24 ) .Find the total volume of water pumped in one day.

(a)

To determine

To calculate: The average hourly flow rate from midnight to noon 0t12, when flow rate of water at pumping station during a day in thousands of gallons per hour is modelled as R=53+7sin(πt6+3.6)+9cos(πt12+8.9) for 0t24.

Explanation

Given Information:

The flow rate of water at pumping station during a day in thousands of gallons per hour is given as R=53+7sin(πt6+3.6)+9cos(πt12+8.9) for 0t24.

Formula used:

The formula for average value of a provided function f(x) for [a,b].

Average value=1baabf(x)dx

The formula for definite integral of abcosnudu.

abcosnudu=[sinnun]ab

The formula for definite integral of absinnudu.

absinnudu=[cosnun]ab

The formula for integral of abdu.

abdu=[u]ab

Calculation:

The flow rate of water at pumping station is,

R=53+7sin(πt6+3.6)+9cos(πt12+8.9)

The average hourly flow rate from midnight to noon 0t12 as,

R=1120012[53+7sin(πt6+3.6)+9cos(πt12+8.9)]dt=112012[53+7sin(πt6+3

(b)

To determine

To calculate: The average hourly flow rate from noon to midnight 12t24, when flow rate of water at pumping station during a day in thousands of gallons per hour is given as R=53+7sin(πt6+3.6)+9cos(πt12+8.9) for 0t24.

(c)

To determine

To calculate: The average hourly flow rate for whole day 0t24, when flow rate of water at pumping station during a day in thousands of gallons per hour is given as R=53+7sin(πt6+3.6)+9cos(πt12+8.9) for 0t24.

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