The spread of an invasive plant in a national park is modeled by the logistic…
MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations
Problem
The spread of an invasive plant in a national park is modeled by the logistic differential equation
dtdA=0.04A(1000−A),
where A(t) is the area, in hectares, covered by the plant t months after introduction.
At t=18 months, four separate regions of the park have plant-covered areas listed in the table below.
Region | Area (ha)
A | 100
B | 200
C | 500
D | 1200
(a) Calculate the instantaneous growth rate, in hectares per month, of Region B at t=18 months.
(b) Using numerical values of the differential equation, rank Regions A, B, and C from fastest to slowest instantaneous growth rate at t=18 months. Show your work.
(c) Determine whether the plant-covered area in Region D is increasing or decreasing at t=18 months. Explain your reasoning in the context of the model.