The spread of a rumor through a small college campus is modeled by the differential…

MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations

Problem

The spread of a rumor through a small college campus is modeled by the differential equation

\frac{dR}{dt}=cR\bigl(M-R\bigr),

where

R(t)

is the number of students who have heard the rumor

t

hours after it begins to circulate, and

c

and

M

are positive constants. Campus officials collect the following information: • The rate of spread attains its maximum value of

100

students per hour when

R=500

. • Two hours after the rumor begins,

R=100

and

\dfrac{dR}{dt}=40

. (a) Show analytically that the rate of spread

\dfrac{dR}{dt}

reaches an extreme value when

R=\dfrac{M}{2}

. Classify this extreme as a maximum or a minimum. (b) Use the first bullet point together with your result from part (a) to determine the numerical value of

M

. (c) Employ the second bullet point to calculate the constant

c

. Clearly state any algebraic steps used. (d) Interpret the value of

M

within the context of the rumor-spreading scenario.

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