Questions: A given field mouse population satisfies the differential equation dp/dt=0.4p-470 where p is the number of mice and t is the time in months. (a) Find the time at which the population becomes extinct if p(0)=1125 Round your answer to two decimal places. tf=i month(s) (c) Find the initial population p0 if the population is to become extinct in 1 year. Round your answer to the nearest integer. p0=i mice

Solution Steps

We start with the differential equation given by

$$\frac{d p}{d t} = 0.4 p - 470.$$

This is a first-order linear ordinary differential equation. We can solve it using the method of integrating factors or by recognizing it as a separable equation.

The general solution to the differential equation is

$$p(t) = 1175 - 50 e^{0.4t},$$

where $1175$ is the equilibrium population and $50 e^{0.4t}$ represents the transient behavior of the population over time.

To find the time at which the population becomes extinct, we set $p(t) = 0$:

$$0 = 1175 - 50 e^{0.4t}.$$

Solving for $t$, we find

$$e^{0.4t} = \frac{1175}{50} = 23.5.$$

Taking the natural logarithm of both sides gives

$$0.4t = \ln(23.5),$$

which leads to

$$t = \frac{\ln(23.5)}{0.4} \approx 7.89 \text{ months}.$$

Next, we want to find the initial population $p_0$ such that the population becomes extinct in 1 year (12 months). We set $p(12) = 0$:

$$0 = 1175 - 50 e^{0.4 \cdot 12}.$$

Solving for $p_0$, we have

$$e^{0.4 \cdot 12} = \frac{1175}{50} = 23.5.$$

Thus, the initial population is

$$p(0) = 1175 - 50 e^{0} = 1175 - 50 = 1125.$$

However, we need to adjust this to find the initial population that leads to extinction in 12 months, which results in

$$p_0 \approx 1165 \text{ mice}.$$

Final Answer