Question

33\. Assume that after the flood the human population was 6 and that 200 years later the population was \(1,000,000\). Find the annual rate of growth of the population (Euler). (Hint: If the annual rate of growth is \(1 / x\), then the equation for the problem is $$ \left.6\left(\frac{1+x}{x}\right)^{200}=1,000,000 .\right) $$

Short answer

Answer: The approximate annual rate of growth is 473.62.

Step-by-step solution

The equation is already simplified, and it's given as: $$ 6\left(\frac{1+x}{x}\right)^{200}=1,000,000 . $$ #Step 2: Isolate the term with x#

Divide both sides of the equation by 6 to get: $$ \left(\frac{1+x}{x}\right)^{200}=\frac{1,000,000}{6} $$ Now, we have: $$ \left(\frac{1+x}{x}\right)^{200}=166,666.\overline{6} $$ #Step 3: Apply the 200th root to both sides#

To remove the exponent of 200, we will apply the 200th root to both sides: $$ \sqrt[200]{\left(\frac{1+x}{x}\right)^{200}}=\sqrt[200]{166,666.\overline{6}} $$ Now, we have: $$ \frac{1+x}{x}=\sqrt[200]{166,666.\overline{6}} $$ #Step 4: Solve for x#

To isolate x, we can rewrite the equation as follows: $$ 1+x = x\cdot\sqrt[200]{166,666.\overline{6}} $$ Now, we subtract x from both sides: $$ 1= x(\sqrt[200]{166,666.\overline{6}}-1) $$ Finally, we divide by \(\sqrt[200]{166,666.\overline{6}} - 1\): $$ x = \frac{1}{\sqrt[200]{166,666.\overline{6}} - 1} $$ To find the value of x, we can approximate the 200th root of 166,666.\overline{6} using a calculator and then calculate the value of x. In scientific notation, we get approximately x ≈ 2.11×10⁻³ and the annual rate of growth is 1/x ≈ 473.62.

Key concepts

Exponential Growth

Exponential growth is a process that increases quantity over time at a rate proportional to its current value. It is one of the most important concepts in understanding how populations change. In this type of growth, the population grows by a constant percentage rather than a constant amount, resulting in a rapid increase in numbers as time progresses.

The equation that typically represents exponential growth is: \[ P(t) = P_0e^{rt} \] where:

• \( P(t) \) is the population at time \( t \),
• \( P_0 \) is the initial population,
• \( r \) is the growth rate, and
• \( e \) is the base of the natural logarithm (approximately equal to 2.71828).

In the context of the textbook problem, the growth isn't continuous but happens annually, and hence the formula takes on a discrete form involving a base larger than one, raised to the power of the number of time periods. This results in the population at a future time being a multiple of the population at the present or past time.

Mathematical Modeling

Mathematical modeling involves creating equations to describe real-world phenomena. When we look at population dynamics, such models help us predict future population sizes based on current and past data.

In the exercise, we use a mathematical model to determine the annual growth rate of a human population after a flood. We are given a beginning population and its size after a certain period, and we need to model this situation with an equation that allows us to calculate the annual rate of growth. These mathematical models rely on assumptions, such as consistent growth rate in this case, and allow us to make predictions about future population numbers. An important aspect of mathematical models is their ability to adjust to new information, which can then provide more accurate predictions.

Euler's Method for Population Growth

Euler's Method is a numerical technique often used for solving ordinary differential equations, which can be applied to estimate population growth over time. While the original context of Euler's Method deals with continuous growth and the differential equations that model such a process, in our textbook problem, we are using a discrete variant that's associated with Euler's name.

The process essentially 'steps through' time in regular intervals, recalculating population size based on the growth rate at each step. In the exercise, we can simplify the concept and apply a form of Euler's method by considering the growth factor as \( (1+x)/x \), with our intervals being years. After following these steps and solving for \( x \), we identify the consistent annual multiplier that models the population growth from 6 to 1,000,000 over 200 years.

The understanding and application of Euler's Method in discrete contexts, such as the textbook problem, empower students to grapple with complex systems that can be broken down into steps and understood in a piecewise fashion, ultimately leading to a solution that would otherwise be difficult to attain in an elegant analytical form.