The rate of change, dp/dt, of the number of people at a dance who have heard a rumor is modeled by a logistic differential equation. There are 2000 people at the dance. At 9pm, the numver of people who have heard the rumor is 400

(dP/dt) = 0.00078P (2000 - P)

Step-by-step explanation:

A logistic differential equation is given as

(dy/dx) = ay(b - y)

where a and b are constants.

So, for this question,

(dP/dt) = aP(b - P)

b is the total possible value that P can attain and for this question, b = 2000 (the total number of people at the party)

Taking the time at 9pm to be t = 0,

At t = 0,

P(0) = 400 and (dP/dt) = 500 at t = 0

Inserting these into the differential equation

(dP/dt) = aP(b - P)

500 = 400a(2000 - 400)

400a = (500/1600)

400a = 0.3125

a = (0.3125/400)

a = 0.00078125

(dP/dt) = aP(b - P)

Inserting the constants

(dP/dt) = 0.00078125P (2000 - P)

Hope this Helps!!!