Q. 65 - Calculus

Question

Suppose that the current population of Freedonia is$1.08$ million people and that the continuous growth rate of the population is $1.39\%$.

(a) Set up a differential equation describing $\frac{dP}{dt}$ , and solve it to get a formula for the population $P\left(t\right)$ of Freedonia $t$ years from now.

(b) According to your model, how long ago was it that the population of Freedonia was just half a million people? 

(c) How long will it take for the population of Freedonia to double from its current size? 

Step-by-Step Solution

Answer

Part $\left(a\right)$: The formula for the population $P\left(t\right)$ of Freedonia $t$ years from now is,$P\left(t\right)=1.08e^{0.0139t}$

Part $\left(b\right)$: Population of the country was half a million between $55$ and $56$ years from now.

Part $\left(c\right)$: Under $50$ years, country's population will become double of the current population.

Part a Step 1. Given information

Population of Freedonia is $1.08$ million

Continuous growth rate of the population is $1.39\%$ . 

Part a Step 2. Forming the equations

Let us assume that the current population is P of a country is $1.08$ million people and the continuous growth rate of the population is $1.39\%$.

 If $P\left(t\right)$ is the population of the country at any time $t$, then according to the given information the growth rate is $1.39\%$ of the population. So, the growth rate is $0.0139$ times the population. This implies that the growth rate of population is equal to $0.0139P$.

$\frac{dP}{dt}=0.0139P$

 Given that the present population of the country is $1.08$ million. So this implies at $t=0$; then it implies that $P\left(0\right)=1.08$.

Part a Step 3. Integrate the obtained equation on both sides.

$$\begin{gathered} \frac{dp}{dt}=0.0139P \\ \frac{dp}{P}=0.0139dt \\ \int \frac{dp}{P}=\int 0.0139dt \\ \ln \left(P\right)=0.0139t+c \\ P=e^{0.0139t+c} \\ P=V{e}^{0.0139t} \\ at t=0 P= 1.08 \\ V= 1.08 \\ P\left(t\right)=1.08e^{0.0139t} \end{gathered}$$

Part b Step 1 . Time when half the population

The current population of the country is $1.08$ million. Suppose that the population was half a million $t$ years back from now. This means that in $t$years the population has grown from half a million to the current population of $1.08$ million  and we have to find t. So, take $P\left(t\right)=1.08$ and

 $P\left(0\right)=0.5$ in the solution to get

$$\begin{gathered} 1.08=0.5e^{0.0139t} \\ {e}^{0.0139t}=\frac{1.08}{0.5} \\ t=55.4 \end{gathered}$$

Population of the country was half a million between $55$ and $56$ years from now.

Part c Step 1 . Time to double the population

The current population of the country is $1.08$ million. Suppose that it takes $t$ years from now for the population to get doubled, that is, it will become $2.16$ million. Substitute $P\left(t\right)=2.16$ in the solution obtained in part $\left(a\right)$ to get

 $$\begin{gathered} 2.16=1.08e^{0.0139t} \\ {e}^{0.0139t}=2 \\ t=\frac{1}{0.0139}\ln 2 \\ =49.87 \end{gathered}$$

Under $50$ years, country's population will become double of the current

population.