The flow rate for water at a pumping station on a given day is
[tex]R(t)=-t^2+24 t+100[/tex]
([tex]R[/tex] is the flow rate in thousands of gallons/hour and [tex]t[/tex] is in hours [tex][0,24][/tex])
Find the average flow rate during the course of the day.
Answer (1)
Set up the integral for the average flow rate: 24 − 0 1 ∫ 0 24 ( − t 2 + 24 t + 100 ) d t .
Evaluate the definite integral: ∫ 0 24 ( − t 2 + 24 t + 100 ) d t = 4704 .
Divide the result by the length of the interval: 24 4704 = 196 .
The average flow rate during the course of the day is 196 thousand gallons/hour.
Explanation
Problem Setup We are given the flow rate function R ( t ) = − t 2 + 24 t + 100 , which represents the flow rate of water at a pumping station in thousands of gallons per hour, where t is the time in hours over the interval [ 0 , 24 ] . We want to find the average flow rate during the course of the day.
Average Flow Rate Formula To find the average flow rate, we need to calculate the definite integral of the flow rate function over the given interval and then divide by the length of the interval. The formula for the average flow rate is:
Average flow rate = b − a 1 ∫ a b R ( t ) d t
where a = 0 and b = 24 .
Substituting the Function Substituting the given function R ( t ) into the formula, we get:
Average flow rate = 24 − 0 1 ∫ 0 24 ( − t 2 + 24 t + 100 ) d t
Evaluating the Integral Now, we need to evaluate the definite integral:
∫ 0 24 ( − t 2 + 24 t + 100 ) d t = [ − 3 1 t 3 + 12 t 2 + 100 t ] 0 24
Plugging in Limits Plugging in the limits of integration, we have:
[ − 3 1 ( 24 ) 3 + 12 ( 24 ) 2 + 100 ( 24 )] − [ − 3 1 ( 0 ) 3 + 12 ( 0 ) 2 + 100 ( 0 )]
= [ − 3 1 ( 13824 ) + 12 ( 576 ) + 2400 ] − 0
= − 4608 + 6912 + 2400 = 4704
Calculating Average Flow Rate Now, we divide the result by the length of the interval (24):
Average flow rate = 24 4704 = 196
Final Answer Therefore, the average flow rate during the course of the day is 196 thousand gallons/hour.
196
Examples
Understanding average flow rates is crucial in many real-world applications. For instance, consider a water reservoir supplying a city. By calculating the average flow rate of water entering the reservoir over a day, engineers can effectively manage water resources, ensuring a consistent supply for the city's needs. This calculation helps in planning water distribution, predicting potential shortages, and optimizing the pumping schedules to maintain adequate water levels. It's a practical application of calculus that directly impacts our daily lives.
