Problem 15 - Calculus

Question

A town has a population of 1000 people at time \(t=0\) In each of the following cases, write a formula for the population, \(P,\) of the town as a function of year \(t\) (a) The population increases by 50 people a year. (b) The population increases by \(5 \%\) a year.

Step-by-Step Solution

Answer

(a) \(P(t) = 1000 + 50t\); (b) \(P(t) = 1000(1.05)^t\).

Step 1: Identifying Population Growth Type for Case (a)

In this case, the population increases by a fixed number of people every year. This is a case of arithmetic or linear growth. The constant increase is 50 people per year.

Step 2: Formulating the Linear Growth Equation

For linear growth, the population at any year can be expressed as \(P(t) = P_0 + rt\), where \(P_0\) is the initial population, \(r\) is the rate of increase, and \(t\) is the time in years. Here, \(P_0 = 1000\) and \(r = 50\). Thus, the formula becomes \(P(t) = 1000 + 50t\).

Step 3: Identifying Population Growth Type for Case (b)

In this case, the population increases by a percentage each year. This is exponential growth because the population grows by a certain percentage of the current population each year.

Step 4: Formulating the Exponential Growth Equation

For exponential growth, the population formula is \(P(t) = P_0(1+r)^t\), where \(P_0\) is the initial population, \(r\) is the rate of increase expressed as a decimal, and \(t\) is the time in years. Here, \(P_0 = 1000\) and \(r = 0.05\) (which is 5% expressed as a decimal). Thus, the formula becomes \(P(t) = 1000(1.05)^t\).

Key Concepts

Arithmetic Growth

Arithmetic growth, also known as linear growth, occurs when a population increases by a fixed number each year. Imagine the population of a town growing by exactly 50 people every year. This straight-line growth is easy to predict, as each year's population depends directly on the previous year's population. For arithmetic growth, we use a simple equation to represent the population over time: \[ P(t) = P_0 + rt \]Where:

• \( P(t) \) is the population at year \( t \)
• \( P_0 \) is the initial population
• \( r \) is the constant increase per year

In our example, \( P_0 \) starts at 1000 people and \( r \) equals 50 people per year. Thus, the equation becomes \( P(t) = 1000 + 50t \). This means if you fast forward 5 years, the town will have 1250 people. Arithmetic growth is simple and predictable, making it useful for straightforward scenarios with fixed additions.

Exponential Growth

Exponential growth is different from arithmetic growth because the population increases by a fixed percentage, not a fixed number. This creates a curve of growth that accelerates over time. Let's consider a town's population growing by 5% each year. This means that each year's growth builds on the previous year's total, creating a multiplier effect. The formula for exponential growth is: \[ P(t) = P_0(1 + r)^t \]Where:

• \( P(t) \) is the population at year \( t \)
• \( P_0 \) is the initial population
• \( r \) is the growth rate as a decimal

In our scenario, \( P_0 = 1000 \) and \( r = 0.05 \). Therefore, the equation is \( P(t) = 1000(1.05)^t \). If you look 5 years into the future, the population would be approximately 1276 people, demonstrating how much faster growth can happen compared to arithmetic models. Exponential growth commonly appears in finance, population studies, and other areas where growth compounds over time.

Mathematical Modeling

Mathematical modeling plays a critical role in understanding population growth. A model uses mathematical equations to simulate real-world scenarios. By representing population growth mathematically, we can predict future population sizes and make informed decisions or plans based on these predictions. In the population growth scenario:

• Arithmetic models provide a straightforward prediction when increases are fixed.
• Exponential models offer a complex, yet realistic perspective when growth rates compound.

The choice between arithmetic and exponential models depends on the situation. When modeling population growth:

• Use arithmetic models if each year's growth is a set number.
• Use exponential models if the growth is a consistent percentage of the current population.

A good mathematical model reflects real-world behavior over time. Understanding these models helps not just in academics but also in real-world applications like urban planning, ecology, and even economics. Models enable scientists and policy-makers to forecast future conditions and devise strategies that help manage resources efficiently.