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2. A water treatment plant processes gray water at a rate given by r(t) = 16e0.2 million gallons per year, where t is the time measured in years for 0 St≤ 5. Write an expression that gives the amount of gray water processed by the treatment plant during the time interval 0 ≤ t ≤5.

2. A water treatment plant processes gray water at a rate given by r(t) = 16e0.2 million gallons per year, where t is the time measured in years for 0 St≤ 5. Write an expression that gives the amount of gray water processed by the treatment plant during the time interval 0 ≤ t ≤5.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Calculus Applications of Integration

rates using integration

### Calculus and Integrals: Real-World Applications ### Problem 4: Rate of Fans Entering a Stadium **Problem Statement:** Football fans enter a sports stadium at a rate of \( r(t) \) people per hour, where \( t \) is time in hours. If there are 18,000 people in the stadium at the end of the first hour, write an expression that would represent the number of fans inside the stadium by the end of the second hour. **Solution Approach:** To find the number of fans inside the stadium by the end of the second hour, integrate the rate function \( r(t) \) over the time interval from 1 hour to 2 hours. Add this to the initial number of fans (18,000) already in the stadium at the end of the first hour. ### Problem 2: Water Treatment Plant Processing Rate **Problem Statement:** A water treatment plant processes gray water at a rate given by \( r(t) = 16e^{0.2t} \) million gallons per year, where \( t \) is the time measured in years for \( 0 \leq t \leq 5 \). Write an expression that gives the amount of gray water processed by the treatment plant during the time interval \( 0 \leq t \leq 5 \). **Solution Approach:** To determine the total amount of gray water processed over the given time interval, integrate the rate function \( r(t) \) from \( t = 0 \) to \( t = 5 \) years. This will provide the total volume of gray water processed by the plant during this period. --- #### Detailed Explanation of Diagrams and Graphs: In both problem statements above, consider plotting the rate functions \( r(t) \) against time to visualize the functional change over the given intervals. For Problem 2 specifically: - **Graph of \( r(t) = 16e^{0.2t} \)** - **Y-axis:** Rate of gray water processing in million gallons per year. - **X-axis:** Time \( t \) in years. - **Curve:** Exponential curve starting from \( r(0) = 16 \) at \( t = 0 \) and increasing as \( t \) grows, reflecting the exponential increase in the rate of water processing. Such visualizations can aid in understanding how the amount processed
Transcribed Image Text:### Calculus and Integrals: Real-World Applications ### Problem 4: Rate of Fans Entering a Stadium **Problem Statement:** Football fans enter a sports stadium at a rate of \( r(t) \) people per hour, where \( t \) is time in hours. If there are 18,000 people in the stadium at the end of the first hour, write an expression that would represent the number of fans inside the stadium by the end of the second hour. **Solution Approach:** To find the number of fans inside the stadium by the end of the second hour, integrate the rate function \( r(t) \) over the time interval from 1 hour to 2 hours. Add this to the initial number of fans (18,000) already in the stadium at the end of the first hour. ### Problem 2: Water Treatment Plant Processing Rate **Problem Statement:** A water treatment plant processes gray water at a rate given by \( r(t) = 16e^{0.2t} \) million gallons per year, where \( t \) is the time measured in years for \( 0 \leq t \leq 5 \). Write an expression that gives the amount of gray water processed by the treatment plant during the time interval \( 0 \leq t \leq 5 \). **Solution Approach:** To determine the total amount of gray water processed over the given time interval, integrate the rate function \( r(t) \) from \( t = 0 \) to \( t = 5 \) years. This will provide the total volume of gray water processed by the plant during this period. --- #### Detailed Explanation of Diagrams and Graphs: In both problem statements above, consider plotting the rate functions \( r(t) \) against time to visualize the functional change over the given intervals. For Problem 2 specifically: - **Graph of \( r(t) = 16e^{0.2t} \)** - **Y-axis:** Rate of gray water processing in million gallons per year. - **X-axis:** Time \( t \) in years. - **Curve:** Exponential curve starting from \( r(0) = 16 \) at \( t = 0 \) and increasing as \( t \) grows, reflecting the exponential increase in the rate of water processing. Such visualizations can aid in understanding how the amount processed
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