The number of students in a cafeteria is modeled by the function p that satisfies the logistic differential equation dp/dt = 1/2

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2 months ago

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The number of students in a cafeteria is modeled by the function p that satisfies the logistic differential equation dp/dt = 1/2

000 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?

Physics

2 answers:

Softa [3K] · Answer 1 · 2026-02-17T20:42:22+03:00

The function provided describes how the number of students changes over time. With p representing the number of students, taking the derivative with respect to time gives the rate of change. We aim to find the highest value (maximum) of this rate. Normally, this involves finding where the derivative of the rate equals zero, but since we don't have p(t) explicitly, solving the differential equation entirely would be complex.
Instead, because the rate function is a parabola, its maximum is at the vertex, which we can calculate directly.
$p'(t)=\frac{1}{2000}p(200-p)\\ p'(t)=\frac{1}{2000}(200p-p^2)\\ p'(t)=-\frac{p^2}{2000}+\frac{p}{10}$
Given the parabola parameters:
$a=-\frac{1}{2000}\\ b=\frac{1}{10}\\ c=0$
The vertex (p-coordinate) is found using:
$p_v=\frac{-b}{2a}\\ p_v=100$
Plugging this into the original function provides the maximum rate:
$p'_{max}=-\frac{100^2}{2000}+\frac{100}{10}=5$students/s$
For a graph of this derivative, see: https://www.desmos.com/calculator/tgmnxqb7fd