Question

The number of students in a cafeteria is modeled by the function p that satisfies the logistic differential equation dp/dt = 1/2000 p(200-p), where t is the time in seconds and p(0) = 25. what is the greatest rate of change, in students per second, of the number of students in the cafeteria?

Answer 1

Answer:

Maximum rate of change will be

$(\frac{dp}{dt})_{max} = 5$ at p = 100

Explanation:

As we know that rate of change in students per second is given by

$r = \frac{dp}{dt} = \frac{1}{2000} p(200 - p)$

here we need to find the greatest rate of change in the number of students

So in order to find the greatest value of the rate of students we have to put the differentiation of of the above function to be zero

so we have

$\frac{dr}{dp} = 0$

$0 = \frac{1}{2000} (200 - 2p)$

by solving above equation we have

$p = 100$

so maximum rate of students will be at the condition when p = 100

so the value of maximum rate will be

$\frac{dp}{dt} = \frac{1}{2000}(100)(200 - 100)$

$\frac{dp}{dt} = 5$