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 models the rate of change of the fraction $p$ of the population that 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 the fraction $p$ approaches 1 as $t$ increases, which means eventually the whole population will have heard the news story. ‖ ‖ step 4 ⋮ The term $kp(t)$ represents the rate of increase of $p$ due to the people who have heard the news, while $p'(t)$ is the actual rate of change of $p$. ‖ ‖ step 5 ⋮ The correct form of the differential equation that describes the situation is $\frac{dp}{dt} = k(1 - p)$. ‖ ***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 $(1 - p)$.⚹◊What is the solution to the differential equation $\frac{dp}{dt}=kp(1-p)$ describing the population's awareness of a breaking news story?⍭ Generate me a similar question◊

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