Explanation Theorem used: Separation of Variables: The solution of the differential equation f ( y ) y ′ = g ( x ) is given implicitly by the equation, ∫ f ( y ) d y = ∫ g ( x ) d x . Formula used: Unlimited growth model: d y d t = k y , k > 0 , t > 0 Calculation: It is given that chlorine based pool disinfectant contains an active ingredient having a half-life of 4 days. Let y represent the active ingredient in the pool. Let the initial active ingredient be y 0 . Therefore, y ( 0 ) = y 0 . After 4 days, the active ingredient reduces to the half of the initial value. That is, y ( 4 ) = 1 2 y 0 . The rate of change of active ingredients is proportional to the active ingredients in the pool. That is, d y d t = k y , k > 0 , t > 0 . Rewrite the differential equation d y d t = k y as, d y d t = k y d y y = k d t 1 y d y = k d t Comparing the equation 1 y d y = k d t and f ( y ) y ′ = g ( x ) , it is observed that f ( y ) = 1 y and g ( t ) = k . By the above theorem, the solution is given by, ∫ f ( y ) d y = ∫ g ( t ) d t ∫ 1 y d y = ∫ k d t ∫ ( 1 y ) d y = ∫ k d t ln y = k t + C 1 Simplify further as, e ln ( y ) = e k t + C 1 y = e k t + C 1 y = C e k t , where C = e C 1 Apply the initial condition, y ( 0 ) = y 0 and y ( 4 ) = 1 2 y 0

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Chapter 9.2, Problem 64E

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To find: How often the disinfectant is to be added to a pool so that at least 30% of the active ingredient is present.

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Chapter 9.2, Problem 63E

Chapter 9.2, Problem 65E

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How often the disinfectant is to be added to a pool so that at least 30% of the active ingredient is present.

Solution Summary: The author explains how the disinfectant is to be added to a pool once in 7 days so that at least 30% of the active ingredient is present.

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To find: How often the disinfectant is to be added to a pool so that at least 30% of the active ingredient is present.

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