Question

A population of insects grows at a rate proportional to its size ((P)). Write a differential equation for the size of the population ((P)) as a function of time ((t)), assuming (k) is the constant of proportionality.

a. (dP/dt = kP)
b. (dP/dt = k/P)
c. (dP/dt = k)
d. (dP/dt = kP^2)
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Answer 1

Correct option is b). The correct differential equation for the population growth that is proportional to its size is dP/dt = kP, representing exponential growth.

Step-by-step explanation:

The differential equation that captures the growth of a population of insects at a rate proportional to its size (P) as a function of time (t) would be dP/dt = kP, where 'dP/dt' represents the rate of change of the population over time, and 'k' is the constant of proportionality. This equation suggests that as the population increases, the growth rate increases proportionally, which is a characteristic of exponential growth. Therefore, the correct answer to the student's question is option a: dP/dt = kP.