A colony of plants is modeled by the differential equation

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

Problem

\dfrac{dy}{dt}=0.1\,y\,(100-y)

, where

y(t)

is the number of plants and

t

is measured in days. Determine the initial population

y(0)=y_0

that will cause the population growth rate to attain its maximum value exactly

20

days after observations begin. (1) Obtain a general solution

y(t)

that satisfies the initial condition

y(0)=y_0

. Express your answer in terms of

y_0

. (2) Set up and solve an equation for

y_0

such that the growth rate is maximized at

t=20

days. (3) Provide the numerical value of

y_0

and state whether this value is less than, equal to, or greater than one-half of the carrying capacity.