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) $\frac{d p}{d t}=k p$
(B) $\frac{d p}{d t}=k(1-p)$
(C) $\frac{d p}{d t}=1-k p$
(D) $\frac{d p}{d t}=-k p$

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) $\frac{d p}{d t}=k p$
(B) $\frac{d p}{d t}=k(1-p)$
(C) $\frac{d p}{d t}=1-k p$
(D) $\frac{d p}{d t}=-k p$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ The given differential equation is $\frac{dp}{dt} = k(1 - p)$. This equation represents the rate of change of the fraction $p$ of the population who has heard the news story. ‖ ‖ step 2 ⋮ The solution to this differential equation is $p(t) = c_1 e^{-kt} + 1$, where $c_1$ is a constant determined by initial conditions. ‖ ‖ step 3 ⋮ This is a first-order linear ordinary differential equation, and the solution shows that $p(t)$ approaches 1 as $t$ increases, which means eventually the whole population will have heard the news story. ‖ ‖ step 4 ⋮ The term $kp(t)$ would represent the rate of change of $p$ if the rate were proportional to $p$ itself, which is not the case here. ‖ ‖ step 5 ⋮ The correct equation that describes the relationship is $\frac{dp}{dt} = k(1 - p)$. ‖ ***B*** ∻Key Concept∻ ⚹First-order linear ordinary differential equation⚹ ∻Explanation∻ ⚹The rate of change of a quantity is proportional to the difference between a constant and that quantity, which is represented by a first-order linear ordinary differential equation. In this case, the rate at which the news spreads is proportional to the fraction of the population that has not yet heard it, $(1 - p)$.⚹◊What is the solution to the differential equation $\frac{dp}{dt} = kp(1-p)$ for the fraction $p$ of the population?⍭ Generate me a similar question◊

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