Problem 14 - Applied Calculus

Question

A population, currently 200 , is growing at \(5 \%\) per year. (a) Write a formula for the population, \(P\), as a function of time, \(t,\) years in the future. (b) Graph \(P\) against \(t\) (c) Estimate the population 10 years from now. (d) Use the graph to estimate the doubling time of the population.

Step-by-Step Solution

Answer

(a) \(P(t) = 200 (1.05)^t\). (b) Graph is exponential. (c) Approximately 327. (d) Doubling time is about 14 years.

Step 1: Establish the initial population and growth rate

The initial population is given as 200, and it is growing at a rate of \(5\%\) per year. We express this growth rate as a decimal, which is \(0.05\).

Step 2: Develop the population growth formula

The formula for population growth is given by the equation for exponential growth: \(P(t) = P_0 (1 + r)^t\), where \(P_0\) is the initial population, \(r\) is the growth rate, and \(t\) is the time in years. For this problem, the equation becomes \(P(t) = 200 (1 + 0.05)^t\).

Step 3: Graph the population function against time

To graph \(P(t) = 200 (1 + 0.05)^t\), you will plot \(t\) on the x-axis and \(P(t)\) on the y-axis. You can calculate and plot several points such as at \(t=0\), \(t=5\), \(t=10\), etc., to see the curve and note that the graph will exhibit exponential growth.

Step 4: Calculate the population after 10 years

Using the formula \(P(t) = 200 (1 + 0.05)^t\) and substituting \(t = 10\), we calculate: \[ P(10) = 200 (1.05)^{10} \]. This results in \(P(10) \approx 326.59\). Thus, the estimated population in 10 years is approximately 327.

Step 5: Estimate the doubling time using the rule of 70

The doubling time can be estimated by the rule of 70, which is \(t_d = \frac{70}{r}\), where \(r\) is the percent growth rate. Here, \(r = 5\), so \(t_d = \frac{70}{5} = 14\). This means the population will approximately double in 14 years.

Key Concepts

Population Growth

Population growth describes the increase in the number of individuals within a population. It can happen for various reasons:

• Birth rates exceeding death rates
• Immigration into an area surpassing emigration from it

When a population grows at a consistent percentage every year, it exhibits exponential growth. For example, if a population starts with 200 individuals and grows at a rate of 5% annually, it acquires new members based on a consistent increase proportionate to its current size. This results in a rapid increase in population over time.
The formula for exponential population growth is:
\[ P(t) = P_0 (1 + r)^t \]
Here, \(P(t)\) is the population at time \(t\), \(P_0\) is the initial population, and \(r\) is the growth rate expressed as a decimal. Understanding this formula is key to predicting future population sizes.

Doubling Time

Doubling time is the period it takes for a population experiencing exponential growth to double in size. It's a concise way to gauge how quickly a population is expanding.
A helpful tool for estimating doubling time is the 'Rule of 70.' This straightforward method allows you to approximate the doubling time based on the growth rate:
\[ t_d = \frac{70}{r} \]
Here, \(t_d\) is doubling time, and \(r\) is the growth rate percentage. For example, with a growth rate of 5%, the doubling time is calculated as:

• \(t_d = \frac{70}{5} = 14\) years

This means our population of 200 individuals, growing at 5% per year, will double in about 14 years. Doubling time provides a quick snapshot of growth dynamics and is a handy concept in various fields, such as ecology, finance, and any scenario involving exponential growth.

Exponential Function

An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. This function is commonly found in models describing real-world phenomena where the change is proportional to the current value, such as population growth.
The general form of an exponential function is:
\[ f(x) = a (b)^x \]
Where:

• \(a\) is the initial value
• \(b\) is the growth factor (1 plus the growth rate as a decimal)
• \(x\) is the exponent, often representing time

In our population growth example, \(P(t) = 200 (1.05)^t\), the function describes how a population increases exponentially over time. The base \(1.05\) indicates a 5% annual increase, reflecting how growth accumulates on the existing population, leading to a steeper rise as time progresses.
Exponential functions are distinct from linear functions, as they represent growth by constant multiples rather than constant additions, leading to their characteristic J-shaped curve. Whether dealing with populations, investments, or other exponential scenarios, understanding this concept is crucial for predicting and visualizing growth over time.