Question Statement

Thomas Malthus in 1798 proved that the increase in population of a country or a city at a certain time is proportional to the total population of the country at that time: $\frac{dP}{dt}\propto P$

If at present the population of city $A$ is 20 million and after 4 years, it is expected to be 25 million, what would be the population of the city after 12 years?

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Background and Explanation

This problem is based on the Malthusian growth model, which uses a first-order differential equation. It assumes that the rate of growth is directly proportional to the current population, leading to an exponential growth pattern over time.

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Solution

Step 1: Set up the Differential Equation

Based on the problem statement, the rate of change of population $P$ with respect to time $t$ is proportional to $P$: $\frac{dP}{dt}\propto P$ $\frac{dP}{dt}=kP$

To solve this, we use the method of separation of variables: $\frac{dP}{P}=kdt$

Step 2: Integrate to find the General Solution

Integrating both sides of the equation: $\int \frac{dP}{P}=k\int dt$ $\ln P=kt+c$

To isolate $P$, we take the exponential of both sides: $P=e^{kt+c}$ $P=e^c\cdot e^{kt}$ Let $e^c$ be represented by a constant $C$: $P=C{e}^{kt}$

Step 3: Find the Initial Constant ($C$)

We are given that at "present" ($t=0$), the population is 20 million: $20=C{e}^{k(0)}$ $20=C{e}^0$ $20=C$

Substituting $C$ back into equation (1), we get: $P=20e^{kt}$

Step 4: Find the Growth Constant ($k$)

We are given that after 4 years ($t=4$), the population is 25 million ($P=25$): $25=20e^{k(4)}$ $\frac{25}{20}=e^{4k}$ $\frac{5}{4}=e^{4k}$ $1.25=e^{4k}$

Taking the natural logarithm ($\ln$) on both sides to solve for $k$: $\ln (1.25)=\ln (e^{4k})$ $\ln (1.25)=4k\cdot \ln e$ Since $\ln e=1$: $0.2232\approx 4k$ $k=\frac{0.2232}{4}$ $k=0.0558$

Substituting $k$ back into equation (2): $P=20e^{0.0558t}$

Step 5: Calculate Population after 12 Years

To find the population when $t=12$, we substitute the value into equation (3): $P=20e^{(0.0558)(12)}$ $P=20e^{0.6696}$

Using the value $e^{0.6696}\approx 1.9534$: $P=20(1.9534)$ $P=39.068\text{ million}$

Rounding to two decimal places, the population is 39.06 million.

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Key Formulas or Methods Used

• Malthusian Growth Model: $\frac{dP}{dt}=kP$
• Separation of Variables: A technique to solve differential equations by grouping like variables.
• Exponential Growth Equation: $P(t)=P_0{e}^{kt}$
• Logarithmic Properties: $\ln (e^x)=x$

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Summary of Steps

1. Model the Equation: Translate the proportionality statement into a differential equation $\frac{dP}{dt}=kP$.
2. Solve the DE: Integrate the equation to find the general exponential form $P=C{e}^{kt}$.
3. Apply Initial Conditions: Use the population at $t=0$ to find the constant $C$.
4. Determine Growth Rate: Use the population at $t=4$ to solve for the growth constant $k$.
5. Predict Future Value: Substitute $t=12$ into the specific growth equation to find the final population.