A rumor spreads among a population of \(N\) people at a rate proportional to the product of the number of people who have heard the rumor and the number of people who have not heard the rumor. If \(p\) denotes the number of people who have heard the rumor, which of the following differential equations could be used to model this situation with respect to time \(t\), where \(k\) is a positive constant?

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The relationship between two variables \(x\) and \(y\) that are proportional is:

$$y=kx $$

In our question, the first variable is the rate of spread of a rumor. This is simply given by \(\dfrac{dp}{dt}\).

The second variable is the product of the number of people who have heard the rumor and the number of people who have not heard the rumor:

$$ \text{not heard rumor } \cdot \text{ heard rumor} $$ $$ = (N-p)\cdot p$$

Therefore,

$$y=kx $$ $$ \boxed{\frac{dp}{dt}=k(N-p)p} $$