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Question
Math
Posted 3 months ago
The fraction pp of the population who has heard a breaking news story increases at a rate proportional to the fraction of the population who has not yet heard the news story.

Which equation describes this relationship?
Choose 1 answer:
(A) dpdt=kp\frac{d p}{d t}=k p
(B) dpdt=k(1p)\frac{d p}{d t}=k(1-p)
(C) dpdt=1kp\frac{d p}{d t}=1-k p
(D) dpdt=kp\frac{d p}{d t}=-k p
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Answer from Sia
Posted 3 months ago
Solution by Steps
step 2
The solution to this differential equation is p(t)=c1ekt+1p(t) = c_1 e^{-kt} + 1, where c1c_1 is a constant determined by initial conditions
step 3
This is a first-order linear ordinary differential equation, and the solution shows that the fraction pp approaches 1 as tt increases, which means eventually the whole population will have heard the news story
step 4
The term kp(t)kp(t) represents the rate of increase of pp due to the people who have heard the news, while p(t)p'(t) is the actual rate of change of pp
step 5
The correct form of the differential equation that describes the situation is dpdt=k(1p)\frac{dp}{dt} = k(1 - p)
B
Key Concept
Proportional rate of change
Explanation
The rate of change of the fraction of the population that has heard the news is proportional to the fraction that has not yet heard it, which is represented by the term (1p)(1 - p).

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