write and find the general solution of the differential equation that models the verbal statement. evaluate…

Explanation:

Step1: Write the differential equation

Since the rate of change of $P$ is proportional to $P$, we have $\frac{dP}{dt}=kP$.

Step2: Solve the differential equation

Separate variables: $\frac{dP}{P}=kdt$. Integrate both sides: $\int\frac{dP}{P}=\int kdt$. We get $\ln|P| = kt + C$. Exponentiating both sides gives $P = Ce^{kt}$. Using the initial - condition when $t = 0$, $P=4000$. Substituting into $P = Ce^{kt}$, we have $4000 = Ce^{k\times0}$, so $C = 4000$. Then $P = 4000e^{kt}$. When $t = 1$, $P = 3400$. Substitute into $P = 4000e^{kt}$: $3400=4000e^{k\times1}$. Then $e^{k}=\frac{3400}{4000}=0.85$, so $k=\ln(0.85)$. So the general solution is $P = 4000e^{t\ln(0.85)}=4000(0.85)^{t}$.

Step3: Evaluate the solution at $t = 4$

Substitute $t = 4$ into $P = 4000(0.85)^{t}$: $P = 4000\times(0.85)^{4}=4000\times0.52200625 = 2088.025$.

Answer:

$\frac{dP}{dt}=kP$ $P = 4000(0.85)^{t}$ $2088.025$