(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!!!