Question

in a town with 300 residents, we denote by p(t) the number of people that have heard a certain joke t days after it was told for the first time. it is reasonable to assume that the rate of diffusion of the joke in the town is proportional to the number of people that already heard the joke as well as to the difference between the number of residents and the number of people that already heard the joke. denote by b the level of social interaction. (a) assuming that the town is isolated, that is, the joke cannot be communicated to more people than the current residents, select the differential equation that best describes the situation.

Answer 1

The differential equation expressing the spread of a joke in an isolated town with 300 residents, where the spread is proportional to the number of people who know and do not know the joke, is dp/dt = b · p(t) · (300 - p(t)).

Step-by-step explanation:

The differential equation that best describes the situation where the rate of diffusion of a joke within an isolated town of 300 residents is proportional to the number of people who have heard it and to those who have not is dp/dt = b · p(t) · (300 - p(t)).

In this equation, p(t) is the number of people who have heard the joke by day t, dp/dt represents the rate of change of p(t) over time, and b represents the level of social interaction, which affects how quickly the joke spreads. The term (300 - p(t)) accounts for the number of people who have yet to hear the joke, which decreases as more people learn about it.