Explanation Given: The rate at which the active ingredients evaporates is proportional to the amount present. Half of the ingredient evaporates in the first 30 days. The deodorizer becomes ineffective after 90% of the ingredient evaporates. 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: If the rate of change with respect to time t of a quantity y is proportional to the amount present, then y satisfies the differential equation, d y d t = k y , k > 0 , t > 0 . Calculation: Let y represent the amount of active ingredient. The rate at which the active ingredients evaporates is proportional to the amount present. That is, d y d t = k y , k > 0 , t > 0 . Thus, the problem follows an unlimited growth model. Rewrite the differential equation d y d t = k y in the form of equation f ( y ) y ′ = g ( x ) as follows. d y d t = k y d y y = k d t 1 y d y = k d t Compare the equation 1 y d y = k d t with equation f ( y ) y ′ = g ( x ) . It is observed that, f ( y ) = 1 y and g ( t ) = k . By the above mentioned 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 Let the initial active ingredient be y 0 . Therefore, y ( 0 ) = y 0 . Substitute t = 0 and y = y 0 in y = C e k t

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Chapter 9.2, Problem 63E

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The time period within which one of the deodorizers remain effective.

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Chapter 9.2, Problem 62E

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The time period within which one of the deodorizers remain effective.

Solution Summary: Theorem used: Unlimited growth model: if the rate of change with respect to time t of a quantity is proportional to the amount present, then y satisfies the differential equation.

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