The function P (t) gives the number of insects in a population at time t. The birth rate for the insects is 6.7 % per unit of time and the death rate is 2 % per unit of time. Give the rate of change of the population with respect to time in terms of P.

dP/dt = 0.047P

Step-by-step explanation:

- Let P (t) be the number of insects at time t.

- Let the birth rate for the insects be b (t, P), then b (t, P) = 6.7% = 0.067

- Let the death rate for the insects be d (t, P), the d (t, P) = 2% = 0.02

The number of insects between time t and t + Δt is given as

Total birth = bPΔt

Total death = dPΔt

The rate of change of population with respect to time is

ΔP = bPΔt - dPΔt

ΔP = (b - d) PΔt

ΔP/Δt = (b - d) P

Taking the limit as ΔP approaches 0

dP/dt = (b - d) P

Using the values of b and d, we have

dP/dt = (0.067 - 0.02) P = 0.047

dP/dt = 0.047P