Question

wn with 900 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 c 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. dpdt

Answer 1

The differential equation describing the spread of a joke in a town of 900 residents is dp/dt = c × p(t) × (900 - p(t)), where p(t) is the number of people who have heard it after t days, and c represents the level of social interaction.

Step-by-step explanation:

The differential equation that best describes the spread of a joke through a town of 900 residents is given by the rate of change of the population that has heard the joke, denoted as dp/dt.

The rate at which the joke spreads is proportional to the number of people who have already heard it (p(t)) and to those who have not yet heard it, which is 900 - p(t). Given the level of social interaction denoted as c, the differential equation can be written as:

dp/dt = c × p(t) × (900 - p(t))

This equation is similar to the logistic growth model in biology, where the growth rate of a population is dependent on the current population and the carrying capacity of the environment. In this context, the carrying capacity is the total population of the town, and the current population is the number of people who have heard the joke so far.