Explanation Given: “The model for measuring a student’s performance in the class room is d k d t + l k = λ k where k ( t ) is the student’s knowledge after t weeks, l is a constant called the coefficient of learning, and λ is a constant representing the fraction of available time the student spends performing helpful acts that should increase knowledge of the subject. The students enrolled in a beginning Spanish class are given a pretest to determine their knowledge in the subject. The score, coefficient of learning and the fraction helpful acts of student A are 0.1, 0.8 and 0.9 respectively. And the score, coefficient of learning and the fraction helpful acts of student B are 0.4, 0.8 and 0.7 respectively.” Formula used: Integrating factor: The integrating factor of a function f ( x ) is given by I ( x ) = e ∫ f ( x ) d x . Calculation: Consider the case of student A . Given that the model for measuring a student’s performance in the class room is d k d t + l k = λ l . The coefficient of learning and the fraction helpful acts of student A are l = 0.8 and λ = 0.9 respectively. Find the integrating factor of the first order linear equation d k d t + 0.8 k = 0.72 where f ( t ) = 0.8 and g ( t ) = 0.72 . I ( k ) = e ∫ 0.8 d t = e 0.8 t Multiply both sides of the equation by the integrating factor. e 0.8 t d k d t + 0.8 e 0.8 t k = 0.72 e 0.8 t ( e 0.8 t k ) ′ = 0.72 e 0.8 t [ ∵ ( x y ) ′ = x y ′ + x ′ y ] Integrating on both sides, ∫ ( e 0.8 t k ) ′ d t = 0.72 ∫ e 0.8 t d t e 0.8 t k = 0.72 ( e 0.8 t 0.8 ) + C e 0.8 t k = 0.9 e 0.8 t + C k = 0.9 + C e − 0.8 t Since the student’s score is 0.1 in the beginning, substitute k = 0.1 and t = 0 , in the equation k = 0.9 + C e − 0.8 t . 0.1 = 0.9 + C e − 0.8 ( 0 ) − 0.8 = C e 0 C = − 0.8 Therefore, the knowledge level of student A represented by the function k A ( t ) = 0.9 − 0.8 e − 0.8 t . Substitute t = 6 in this equation to find the knowledge of student A after 6 weeks

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Chapter 9.3, Problem 79E

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To predict: The knowledge level of each student after 6 weeks.

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Chapter 9.3, Problem 78E

Chapter 9.3, Problem 80E

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To predict: The knowledge level of each student after 6 weeks.

Solution Summary: The author calculates the knowledge level of each student after 6 weeks, based on the model for measuring a student's performance in the class room, the coefficient of learning, and the tion helpful acts.

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