The fraction p 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) (dp)/(dt)=-kp (B) (dp)/(dt)=1-kp (C) (dp)/(dt)=kp (D) (dp)/(dt)=k(1-p)

Question

The fraction  p 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)  (dp)/(dt)=-kp (B)  (dp)/(dt)=1-kp (C)  (dp)/(dt)=kp (D)  (dp)/(dt)=k(1-p)

Bytelearn · Accepted Answer

Define Variables: Let's define the variables: p = fraction of the population that has heard the news story 1 - p = fraction of the population that has not yet heard the news story (dp)/(dt) = rate of change of p with respect to time k = constant of proportionality  The problem states that the rate at which people hear the news story is proportional to the fraction of the population who has not yet heard it. This can be mathematically expressed as: (dp)/(dt) = k * (1 - p)  This equation means that the rate of change of p is equal to some constant k times the fraction of the population that has not heard the news story (1 - p).Rate Proportional Equation: Now let's match our derived equation with the given choices: (A) (dp)/(dt) = -kp (This implies the fraction hearing the news decreases over time, which is not the case.) (B) (dp)/(dt) = 1 - kp (This does not represent a proportional relationship to the fraction that has not heard the news.) (C) (dp)/(dt) = kp (This implies the rate is proportional to the fraction that has heard the news, not the fraction that hasn't.) (D) (dp)/(dt) = k(1 - p) (This matches our derived equation and correctly represents the relationship described in the problem.)  Therefore, the correct equation is (D) (dp)/(dt) = k(1 - p).

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