A biologist proposes the differential equation dP dt = 0.09 ,P ,(730-P) to model the mass

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

Problem

A biologist proposes the differential equation

\frac{dP}{dt} = 0.09\,P\,(730-P)

to model the mass

P(t)

, measured in grams, of an algal population in a controlled laboratory pond, where

t

is measured in days. (a) Identify the limiting value that

P(t)

approaches as

t \to \infty

and state the units of this quantity. (b) Describe the physical meaning of the constant

0.09

in the context of the algal population. (c) Explain why the expression

730-P

appears in the differential equation, and discuss how its presence affects the behavior of the solution as

P

approaches

730

. (d) The population changes most rapidly when

P

equals a specific value. Determine that value and justify your conclusion without solving the differential equation.