In a seasonal growth model, a periodic function of time is introduced to account for seasonal variations in the rate of growth. Such variations could, for example, be caused by seasonal changes in the availability of food. (a) Find the solution of the seasonal-growth model dP dt = kP cos(rt − φ) P(0) = P0 where k, r, and φ are all positive constants. (b) By graphing the solution for several values of k, r, and φ, explain how the values of k, r, and φ affect the solution. What can you say about limt→∞ P(t)?

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section: Chapter Questions
Problem 18T
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In a seasonal growth model, a periodic function of time is introduced to account for seasonal variations in the
rate of growth. Such variations could, for example, be caused by seasonal changes in the availability of food.
(a) Find the solution of the seasonal-growth model

dP
dt = kP cos(rt − φ) P(0) = P0

where k, r, and φ are all positive constants.
(b) By graphing the solution for several values of k, r, and φ, explain how the values of k, r, and φ affect the
solution. What can you say about limt→∞
P(t)?

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