The concentration of chemical agents in pool water is regulated by health and safety standards to...

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The concentration of chemical agents in pool water is regulated by health and safety standards to ensure safe use of recreational facilities [2]. Chlorine is one of the most common disinfectants added to pool water in order to suppress the growth of pathogenic microorganisms [1]. This project is concerned with the concentration of chlorine in a mid-sized swimming pool during a regular maintenance cycle. A certain swimming pool is 20m long, 8m wide, and 2m deep. When filled, it contains a volume of 84,000 gallons of water. Recommended levels of chlorine concentration range from 1.3 × 10-5 to 2.6 × 10-5 ounces per gallon of water. The pool's regular maintenance schedule involves filtering, cleaning, and refilling the pool at regular time intervals, while adhering to recommended health and safety standards. A particular incident during one of the regular maintenance cycles can be described by a two-step process as follows. 1. Model formulation - Stage 1 (a) Denote the total amount of chlorine in the pool at time t by Q(t) (in ounces). Write down an expression denoting the rate of change of amount of chlorine. (b) Formulate a differential equation modeling the amount of chlorine, Q(t), during Stage 1. Define any new constants, if necessary. (c) Write down the initial condition(s). 2. Separate variables to solve the differential equation modeling Stage 1. In particular: (a) Obtain a general solution to the differential equation (b) Apply the initial condition(s) to find how the amount of chlorine changes in time during Stage 1. (c) Does your answer make sense physically? Does your solution have the properties you expect it to have? (d) Write down any thoughts you have about your work, including what you think went well, what you are unsure of, and what you'd like to have feedback on. 3. Model formulation - Stage 2 (a) Formulate a differential equation modeling the amount of chlorine during Stage 2. Define any new constants, if necessary. (b) Write down appropriate initial condition(s) for Stage 2, in order to couple the problem to that of Stage 1. 4. Use the method of integrating factors to solve the differential equation modeling Stage 2. In particular: (a) Identify an appropriate integrating factor. (b) Obtain a general solution to the differential equation. (c) Apply the initial condition(s) to find how the amount of chlorine changes in time during Stage 2. (d) Does your answer make sense physically? Does your solution have the properties you expect it to have, in comparison to that of Stage 1? (e) How long does it take for the concentration of chlorine to reach the recommended minimum value of 1.3 x 10-5 ounces per gallon of water? (f) Write down any thoughts you have about your work, including what you think went well, what you are unsure of, and what you'd like to have feedback on. 5. Are there any areas you could improve on? The concentration of chemical agents in pool water is regulated by health and safety standards to ensure safe use of recreational facilities [2]. Chlorine is one of the most common disinfectants added to pool water in order to suppress the growth of pathogenic microorganisms [1]. This project is concerned with the concentration of chlorine in a mid-sized swimming pool during a regular maintenance cycle. A certain swimming pool is 20m long, 8m wide, and 2m deep. When filled, it contains a volume of 84,000 gallons of water. Recommended levels of chlorine concentration range from 1.3 × 10-5 to 2.6 × 10-5 ounces per gallon of water. The pool's regular maintenance schedule involves filtering, cleaning, and refilling the pool at regular time intervals, while adhering to recommended health and safety standards. A particular incident during one of the regular maintenance cycles can be described by a two-step process as follows. 1. Model formulation - Stage 1 (a) Denote the total amount of chlorine in the pool at time t by Q(t) (in ounces). Write down an expression denoting the rate of change of amount of chlorine. (b) Formulate a differential equation modeling the amount of chlorine, Q(t), during Stage 1. Define any new constants, if necessary. (c) Write down the initial condition(s). 2. Separate variables to solve the differential equation modeling Stage 1. In particular: (a) Obtain a general solution to the differential equation (b) Apply the initial condition(s) to find how the amount of chlorine changes in time during Stage 1. (c) Does your answer make sense physically? Does your solution have the properties you expect it to have? (d) Write down any thoughts you have about your work, including what you think went well, what you are unsure of, and what you'd like to have feedback on. 3. Model formulation - Stage 2 (a) Formulate a differential equation modeling the amount of chlorine during Stage 2. Define any new constants, if necessary. (b) Write down appropriate initial condition(s) for Stage 2, in order to couple the problem to that of Stage 1. 4. Use the method of integrating factors to solve the differential equation modeling Stage 2. In particular: (a) Identify an appropriate integrating factor. (b) Obtain a general solution to the differential equation. (c) Apply the initial condition(s) to find how the amount of chlorine changes in time during Stage 2. (d) Does your answer make sense physically? Does your solution have the properties you expect it to have, in comparison to that of Stage 1? (e) How long does it take for the concentration of chlorine to reach the recommended minimum value of 1.3 x 10-5 ounces per gallon of water? (f) Write down any thoughts you have about your work, including what you think went well, what you are unsure of, and what you'd like to have feedback on. 5. Are there any areas you could improve on?

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