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)=k(1-p) (D) (dp)/(dt)=-kp

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)=k(1-p) (D)  (dp)/(dt)=-kp

Bytelearn · Accepted Answer

Define p and rate: Let's denote p as the fraction of the population that has heard the news story at a given time t. The rate at which people hear the news story is given by (dp)/(dt), which is the derivative of p with respect to time t. Since the rate is proportional to the fraction of the population that has not yet heard the news story, we can express this as (dp)/(dt) = k * (1 - p), where k is the constant of proportionality. This is because 1 - p represents the fraction of the population that has not heard the news story. Check given options: Now let's check the given options to see which one matches our expression: (A) (dp)/(dt) = kp does not match because it suggests the rate is proportional to the fraction that has heard the news, not the fraction that hasn't. (B) (dp)/(dt) = 1 - kp does not match because it suggests the rate is equal to 1 minus the product of k and p, which does not represent a proportional relationship. (C) (dp)/(dt) = k(1 - p) matches our derived expression and represents the rate being proportional to the fraction that has not heard the news. (D) (dp)/(dt) = -kp does not match because it suggests the rate is proportional to the fraction that has heard the news and is decreasing, which is not the case described.

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