An insect colony has population insects after days

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

Problem

An insect colony has population P(t)P(t) insects after tt days. The population satisfies the differential equation dPdt=0.05P(1−P2000),P(0)=50.\frac{dP}{dt}=0.05P\Bigl(1-\frac{P}{2000}\Bigr), \quad P(0)=50. Throughout this problem, all populations are measured in insects and all times in days. (a) Show that the solution of the differential equation can be written in the form P(t)=20001+Ae−0.05t,P(t)=\frac{2000}{1+Ae^{-0.05t}}, where AA is a constant to be determined. (b) Use the initial condition to find the exact value of AA. (c) Using your expression for P(t)P(t), determine the exact time required for the population to reach 10001000 insects. Then give a decimal approximation of this time to the nearest tenth of a day. (d) Evaluate lim⁡t→∞P(t)\displaystyle\lim_{t\to\infty}P(t) and interpret its meaning for the insect colony.

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