Explanation Given: r ( t ) = 100 + 72 t − 3 t 2 , where t is the number of hours past midnight. Calculation: Let q ( T ) is the amount of water (in thousands of gallons) consumed in T hours. By fundamental theorem of calculus, we know q ( T ) − q ( 0 ) = ∫ 0 T q ' ( t ) d t = ∫ 0 T r ( t ) d t q ( 0 ) = 0 as day has just started . Since the day has 24 hours, the water consumed by city in a whole day is given by q ( 24 ) = ∫ 0 24 r ( t ) d t = ∫ 0 24 ( 100 + 72 t − 3 t 2 ) d t ∫ 0 24 r ( t ) d t = [ 100 t + 72 t 2 2 − 3 t 3 3 ] 0 24 ∫ 0 24 r ( t ) d t = [ 100 t + 36 t 2 − t 3 ] 0 24 ∫ 0 24 r ( t ) d = [ 100 × 24 + 36 × 24 × 24 − ( 24 ) 3 ] − 0 ∫ 0 24 r ( t ) d = 2400 + 20736 − 13824 ∫ 0 24 r ( t ) d t = 9312 So, the city consumes 9312 thousand gallons of water daily...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 5, Problem 77CRE

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On a typical day, a city consumes water at the rate of $r (t) = 100 + 72 t - 3 t^{2}$ (in thousands of gallons per hour), where $t$ is the number of hours past midnight. What is the daily water consumption? How much water is consumed between 6 PM and midnight?

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Chapter 5, Problem 76CRE

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On a typical day, a city consumes water at the rate of r ( t ) = 100 + 72 t − 3 t 2 (in thousands of gallons per hour), where t is the number of hours past midnight. What is the daily water consumption? How much water is consumed between 6 PM and midnight?

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To find: On a typical day, a city consumes water at the rate of r(t)=100+72t−3t2(in thousands of gallons per hour), where t is the number of hours past midnight. What is the daily water consumption? How much water is consumed between 6 PM and midnight?

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On a typical day, a city consumes water at the rate of $r (t) = 100 + 72 t - 3 t^{2}$ (in thousands of gallons per hour), where $t$ is the number of hours past midnight. What is the daily water consumption? How much water is consumed between 6 PM and midnight?