How to Solve Population Growth Math Problems (With Examples)

Example 1: Modeling Growth in a Small Population

In order to assess India’s rapid population growth, the government decides to investigate a small population of 1000 individuals that appears to have a rate of growth of 2% per year. Using an exponential growth model, help the government determine the size of the small population after 10 years.

Step 1: Set Up the Equation

To solve this problem, we begin by writing a formula for the population growth. We do this using the base equation:

\[P(t)=P_{0}e^{rt}\]

In this equation:

• \(P(t)\) represents the population as a function of time.
• \(P_{0}\) represents the initial population (or the starting population). The initial population is given to be 1000 individuals.
• \(r\) represents the growth constant or annual growth rate. The growth rate is given to be 2%. We write this as a decimal of 0.02 in our formula.
• \(t\) represents the number of years. We are told that the time is 10 years in this problem.

Substituting the information given in the problem into our base equation results in: 

\[P(t)=1000e^{0.02t}\]

Step 2: Calculate the Population After 10 Years

We can now use our exponential equation to evaluate the population after 10 years. To do this, we let \(t=10\) in our equation:

\[ \begin{split} P(t)&=1000e^{0.02t} \\ \\ P(10)&=1000e^{0.02 \cdot 10} \\ \\ P(10)&=1000e^{0.2} \\ \\ P(10)& \approx 1221\\ \\ \end{split}\]

Therefore, after 10 years, the actual population will be approximately 1221 individuals.

Example 2: Solving for Time Using Exponential Growth

Let’s consider a population growth scenario where we need to determine how long it will take for a population to reach a specific size.

A town starts with an original population of 5000, growing at an exponential growth rate of 4% per year. The government is concerned that the population will soon surpass its carrying capacity. Using an exponential growth model, determine how many years it will take for the population to reach 10000 individuals.

Step 1: Write the Exponential Growth Equation

As was the case in example 1, we begin by creating an exponential model for this real-world scenario. We are told that:

• \(P(t)\), the final population, is 10000.
• \(P_{0}\), the initial population (or the starting population), is 5000 individuals.
• \(r\), the rate of growth, is 4%. We will write this as a decimal of 0.04 in our exponential equation.
• \(t\), the amount of time in years, is our unknown.

Substituting these known values into our equation results in: 

\[10000=5000e^{0.04t}\]

Step 2: Solve for the Exponent

Notice that the value of our unknown is located in the exponent. As a result, this problem requires the use of logarithms. 

But first, we begin by isolating the exponential term by dividing both sides of the equation by 5000:

\[ \begin{split} 10000&=5000e^{0.04t}\\ \\ 2&=e^{0.04t} \end{split}\]

Next, take the natural logarithm (or log base e) of both sides. We then use the properties of logarithms to solve for the exponent:

\[ \begin{split} 2&=e^{0.04t} \\ \\ ln(2) &=ln(e^{0.04t}) \\ \\ ln(2) &=0.04t \\ \\ t & = \frac{ln(2)}{0.04} \\ \\ t&\approx 17.33  \end{split}\]

With a final answer of 17.33, we can conclude that it will take approximately 17.33 years for the population to double from 5000 to 10000 individuals at a growth rate of 4%.