Problem 21 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Suppose we know a population grows exponentially; \(P(2)=1000\) and \(P(4)=1300\). Find the growth equation. (Hint: Write \(P=P_{0} e^{k t}\), or some other form of exponential growth. Put in the given information. Since you don't know \(P_{0}\), divide one equation by the other so that the \(P_{0}\) 's cancel.)

Step-by-Step Solution

Answer

The growth equation for the given problem is \( P = \frac{1000}{e^{2k}} e^{kt} \), where \( k = 0.5 * ln(1.3) \).

Step 1: Substitute the given values into the formula

First replace the variables \( t \) and \( P \) in the formula \( P = P_0 e^{kt} \) with the given time (2 and 4) and population (1000 and 1300) respectively to get two equations: \n - For \( t = 2 \) and \( P = 1000 \), we get: \n \( 1000 = P_0 e^{2k} \) ([1]) \n - For \( t = 4 \) and \( P = 1300 \), we get: \n \( 1300 = P_0 e^{4k} \) ([2])

Step 2: Divide one equation by the other to eliminate \(P_{0}\)

Dividing [2] by [1] will result in cancelling out the unknown \( P_0 \), leading to an equation only with \( k \). \n So, \((1300 = P_0 e^{4k}) / (1000 = P_0 e^{2k})\) simplifies to: \n \( \frac{1300}{1000} = \frac{e^{4k}}{e^{2k}} \). \n This leads to \( 1.3 = e^{2k} \).

Step 3: Solve for \( k \)

To solve for \( k \), first take the natural logarithm (ln) of both sides: \n \( ln(1.3) = ln(e^{2k}) \), which simplifies to \( ln(1.3) = 2k \). \n So, \( k = 0.5 * ln(1.3) \).

Step 4: Solve for \( P_{0} \) using the value for \( k \)

Now that we have the value for \( k \), we can solve for \( P_{0} \) by substituting \( k \) back into one of the original equations. Let's substitute \( k \) into the equation [1]: \n \( 1000 = P_0 e^{2k} \), which gives: \n \( P_{0} = \frac{1000}{e^{2k}} \).

Step 5: Write the final exponential growth equation

Now that we have obtained the values for \(k\) and \(P_0\) , we can write the final population growth equation as \(P = P_0 e^{kt} = \). Simply plug in the values for \(P_0\) and \(k\) obtained, to get the exponential growth equation for the given problem.

Key Concepts

Exponential Function

The exponential function is one of the most important and widely used functions in mathematics, particularly when modeling growth processes. It is defined as a function of the form \( f(t) = P_0 e^{kt} \), where:

• \( P_0 \) is the initial quantity or population at time \( t=0 \).
• \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
• \( k \) is the growth rate constant.
• \( t \) represents time.

When dealing with problems such as population growth, it's common to use the exponential function due to its ability to adequately describe how populations grow under ideal conditions—rapidly and without limit.

Natural Logarithm

The natural logarithm is the logarithm to the base \( e \), which is an important mathematical constant approximately equal to 2.71828. It is denoted as \( ln(x) \) and is the inverse operation of exponentiation when referring to this natural base. This means that if \( e^y = x \), then \( ln(x) = y \). In the context of exponential growth, the natural logarithm is often used to solve for the growth rate \( k \) by isolating it from the exponent. This was seen in Step 3 of our original exercise where the equation \( 1.3 = e^{2k} \) was solved by taking the natural logarithm of both sides, yielding \( ln(1.3) = 2k \) and subsequently \( k = 0.5 * ln(1.3) \). Understanding how to work with the natural logarithm is vital for handling equations involving continuous growth processes.

Rate of Population Growth

The rate of population growth refers to how fast a population increases over time. It is often denoted by the variable \( k \) in exponential growth models. In an idealized scenario where resources are unlimited, the growth rate is constant, and the population size increases at a rate proportional to its current size, resulting in exponential growth. This rate can be positive, indicating growth, or negative, suggesting decline. In the example problem, once the value of \( k \) is found, it tells us how quickly the population is growing. For instance, a larger \( k \) means a faster rate of growth. It's crucial to precisely determine this rate to make informed predictions and understand the implications of growth on resources and environment.