(A) 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: Logistic growth model: If the rate of change with respect to time t of a quantity y is proportional to both the amount present and the difference between y and a limiting value M , then y satisfies the differential equation, d y d t = k y ( M − y ) , k > 0 , t > 0 Calculation: Let y represents the number of people who have heard the rumor. The total number of people in the community is 1,000. Thus, the number of people who are unaware of the rumor will be 1 , 000 − y . Rumor is initiated by 5 people and it is spread to over to 10 people after one day. So, after 1 day, 10 people have heard the rumor. The rate is proportional to the product of aware and unaware people, which is d y d t = k y ( 1 , 000 − y ) , k > 0 , t > 0 . Thus, the problem follows a limited growth model. Rewrite the differential equation d y d t = k y ( 1 , 000 − y ) as follows. d y y ( 1 , 000 − y ) = k d t 1 y ( 1 , 000 − y ) d y = k d t The given differential equation ( y − 1 ) − 1 2 y ′ = x ( x − 1 ) 1 2 is in the form of f ( y ) y ′ = g ( x ) where, f ( y ) = 1 y ( 1 , 000 − y ) and g ( t ) = k . By the above theorem, the solution is given by, ∫ f ( y ) d y = ∫ g ( t ) d t ∫ ( 1 y ( 1 , 000 − y ) ) d y = ∫ k d t (1) Rewrite the integral ∫ ( 1 y ( 1 , 000 − y ) ) d y as follows. ∫ ( 1 y ( 1 , 000 − y ) ) d y = ∫ ( 1000 + y − y 1000 y ( 1000 − y ) ) d y = 1 1000 ∫ ( 1 y + 1 1000 − y ) d y = 1 1000 [ ln y − ln ( 1000 − y ) ] ∫ ( 1 y ( 1 , 000 − y ) ) d y = ∫ k d t 1 1000 [ ln y − ln ( 1000 − y ) ] = k t + C 1 ln y − ln ( 1000 − y ) = 1000 ( k t + C 1 ) ln ( y 1000 − y ) = 1000 ( k t + C 1 ) Simplify further as, e ln ( y 1000 − y ) = e 1000 ( k t + C 1 ) y 1000 − y = e 1000 ( k t + C 1 ) 1000 − y y = C e − 1000 k t , where C = e − 1000 C 1 1000 y − 1 = C e − 1000 k t On further simplification, 1000 y = C e − 1000 k t + 1 y 1000 = 1 C e − 1000 k t + 1 y = 1000 C e − 1000 k t + 1 Given y ( 0 ) = 5 (B) To determine To find: The time taken for the rumor to spread around 850 people.

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Author: Stocker

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Chapter 9.2, Problem 83E

(A)

To determine

To find: The number of people who have heard the rumor after 7 days.

(B)

To determine

To find: The time taken for the rumor to spread around 850 people.

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Chapter 9.2, Problem 82E

Chapter 9.3, Problem 1MP

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The number of people who have heard the rumor after 7 days.

Solution Summary: Theorem used: Separation of Variables: The solution of the differential equation is given implicitly by the equation. The rate of change with respect to time t is proportional to both the amount present and the difference