Explanation Given: A body having a temperature 75 ° F , was found in an alley of constant 50 ° F at 1 am. The temperature of the body dropped to 70 ° F at 2 am. 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 . Calculation: Let y represents the temperature of the body. Assume the time 1 am at which the body was found as, t = 0 . The temperature of the freezer is 0 ° F . Thus, the lower limit will be 0 ° F . The rate of change of temperature is proportional to the difference between temperature and the limit 0 ° F . That is, d y d t = k ( y − 0 ) , k > 0 , t > 0 d y d t = k y Rewrite the differential equation d y d t = k y as follows. d y d t = k y d y y = k d t The given differential equation d y d t = k y is in the form of f ( y ) y ′ = g ( x ) where, 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 ln ( y ) = k t + C 1 e ln ( y ) = e k t + C 1 Simplify further as, y = C e k t , where C = e C 1 . Given that, the initial temperature of the body was 75 ° F and declined to 70 ° F in 1 hour, so that y ( 0 ) = 75 and y ( 1 ) = 70

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Chapter 9.2, Problem 79E

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To find: The time of death according to the given information.

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Chapter 9.2, Problem 78E

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The time of death according to the given information.

Solution Summary: The author calculates the time of death according to the given information. The solution of the differential equation is given implicitly by the equation.

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