Problem 1 - Applied Calculus

Question

A town has a population of 1000 . Fill in the table assuming that the town's population grows by (a) 50 people per year (b) \(5 \%\) per year $$ \begin{array}{c|c|c|c|c|c|c|c} \hline \text { Year } & 0 & 1 & 2 & 3 & 4 & \ldots & 10 \\ \hline \text { Population } & 1000 & & & & & \ldots & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Linear growth: up to 1450; Exponential growth: up to 1628.89 by year 10.

Step 1: Fill the Table for Linear Growth

To complete the table under the assumption of linear growth, add 50 people each year to the population. Start with 1000 for year 0, and each subsequent year, the population will be the previous year's population plus 50. - **Year 0:** 1000 - **Year 1:** 1000 + 50 = 1050 - **Year 2:** 1050 + 50 = 1100 - **Year 3:** 1100 + 50 = 1150 - **Year 4:** 1150 + 50 = 1200 - Continue this pattern until year 10: - **Year 10:** 1450

Step 2: Setup for Exponential Growth

For exponential growth, the population increases by 5% each year. This can be calculated using the formula:\[P = P_0 (1 + r)^t\]where:- \(P_0\) is the initial population (1000)- \(r\) is the growth rate (5% or 0.05)- \(t\) is the number of years.

Step 3: Calculate Population for Exponential Growth

Use the exponential growth formula to calculate the population for each year.- **Year 0:** \(P = 1000 \)- **Year 1:** \(P = 1000(1 + 0.05)^1 = 1050\)- **Year 2:** \(P = 1000(1 + 0.05)^2 \approx 1102.5\)- **Year 3:** \(P = 1000(1 + 0.05)^3 \approx 1157.63\)- **Year 4:** \(P = 1000(1 + 0.05)^4 \approx 1215.51\)- Continue this process until year 10: - **Year 10:** \(P \approx 1628.89\)

Key Concepts

Understanding Linear Growth

Linear growth is a straightforward concept where the population increases by the same amount every year. In the given exercise, the town's population grows by 50 people annually. This type of growth is predictable and consistent.

Linear growth can be seen as a straight incline when plotted on a graph since each year adds exactly 50 new people to the total. Here's how it works:

• Start Point: You begin with an initial population, in this case, 1000 people.
• Annual Increase: Add a fixed number, say 50 people, to this starting number every year.
• Calculation: Yearly total = Previous Year's Population + 50.

The pattern is simple:

• Year 1: 1000 + 50 = 1050
• Year 2: 1050 + 50 = 1100
• And so on...
• By Year 10, you reach 1450 people.

Linear growth is easy to predict, making it useful in cases where resources or other factors are steadily increasing.

Diving into Exponential Growth

Exponential growth is more dynamic than linear growth because the increment itself increases each year. This means that instead of adding a constant number of people annually, you add a percentage of the current population, causing a faster increase in numbers.

The town in the exercise grows by 5% annually, which is an example of exponential growth.

• Base Population: You start again with 1000 people.
• Growth Formula: Use the formula \( P = P_0 (1 + r)^t \).
  • \( P_0 \) is the initial population (1000).
  • \( r \) is the growth rate (0.05 for 5%).
  • \( t \) is the number of years.

This formula compounds growth, meaning:

• Year 1: \( P = 1000(1 + 0.05)^1 = 1050 \)
• Year 2: \( P = 1000(1 + 0.05)^2 \approx 1102.5 \)
• Continuing in this fashion until Year 10 gives \( P \approx 1628.89 \).

Exponential growth is powerful for representing rapidly increasing populations or investments. It's essential in biology, finance, and demography.

Calculating Growth Rate

Understanding how to calculate the growth rate is a vital skill when analyzing population changes. The growth rate tells us how fast the population is increasing over a period of time. To compute this rate correctly, knowing whether the growth is linear or exponential is necessary, as each requires a different approach.

For linear growth:

• Objective: Establish the constant amount added every year (e.g., 50).
• Rate Interpretation: Linear growth is simple since it's a direct addition, not a percentage.

For exponential growth:

• Percentage Change: Calculate the rate as a percentage increase annually, like 5% in the example.
• Using Formula: To find this, you'll work with \( P = P_0 (1 + r)^t \) to see how much it's grown.
  • The "r" part of the formula indicates the percentage increase each year.

Whether growing steadily or rapidly, understanding and calculating these rates will allow you to interpret and predict population changes effectively. It’s important in strategic planning and resources management.