Let represent the number of algae cells (in thousands) in a freshwater tank days…

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

Problem

Let

y(t)

represent the number of algae cells (in thousands) in a freshwater tank

t

days after an experimental nutrient is added. The growth of the algae is governed by the differential equation

\frac{dy}{dt}=0.02\,y\bigl(1000-y\bigr).

Answer each of the following. (a) Determine the long-term limiting value of

y(t)

t\to\infty

. Include units in your answer. (b) The initial population is

y(0)=50

. Is the algae population increasing or decreasing at

t=0

? Evaluate

\dfrac{dy}{dt}

t=0

and interpret the sign of the result in context. (c) Suppose the algae population reaches

y=500

on day

D

. Without solving for

y(t)

, decide whether the rate of change

\dfrac{dy}{dt}

t=D

is greater than, less than, or equal to the rate of change you found in part (b). Justify your conclusion analytically.