Question

2. Assume that the growth rate of a population of ants is proportional to the size of the population at each instant of time. Suppose 100 ants are present initially and 230 are present after 3 days.

a. Write a differential equation that models the population of the ants.

b. Solve the differential equation with the initial conditions.

c. What is the population of the ants after 14 days?

Answer 1

(a) The differential equation that would best represent the given is,
dP/dt = kP
(b) Solving the differential equation,
dP/P = kdt
lnP - lnP₀ = kt
Solving for k,
ln(230) - ln(100) = k(3) ; k = 0.2776
(c) Solving for P at t = 14
ln(P) - ln(100) = 0.2776(14) ; P = 4875.99
The population of the ants after 14 days is approximately 4876.