Problem 19 - Calculus for Biology and Medicine

Question

Find the exponential growth equation for a population that doubles in size every unit of time and that has 40 individuals at time 0 .

Step-by-Step Solution

Answer

The exponential growth equation is \( P(t) = 40 \times 2^{t} \).

Step 1: Identify the Exponential Growth Formula

Start by identifying the formula for exponential growth: \( P(t) = P_0 imes e^{kt} \), where \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population, \( e \) is the base of the natural logarithm, and \( k \) is the growth rate.

Step 2: Insert Initial Conditions

Given that the initial population \( P_0 = 40 \), substitute this into the exponential growth formula to get: \( P(t) = 40 imes e^{kt} \).

Step 3: Utilize Doubling Time

The problem states that the population doubles every unit of time, so after 1 unit of time the population is \( 80 \) (double of \( 40 \)). Thus, \( P(1) = 80 \). Substitute \( t = 1 \) and \( P(t) = 80 \) into the equation: \( 80 = 40 imes e^{k} \).

Step 4: Solve for Growth Rate \( k \)

Divide both sides of the equation by \( 40 \) to isolate \( e^{k} \): \( 2 = e^{k} \). To solve for \( k \), take the natural logarithm of both sides: \( k = \ln(2) \).

Step 5: Write the Final Exponential Growth Equation

Substitute \( k = \ln(2) \) into the exponential growth formula: \( P(t) = 40 imes e^{(\ln(2))t} \). This can be simplified using logarithm properties: \( P(t) = 40 imes 2^{t} \), since \( e^{\ln(2)} = 2 \)..

Key Concepts

Doubling Time

Doubling time in exponential growth scenarios tells us how long it takes for a quantity to double. It's a helpful concept that simplifies the understanding of how rapidly something is growing. In this exercise, we know the population doubles every unit of time. This allows us to set the value of the growth parameter for our equation.

The concept of doubling time is applied directly in this exercise when it is stated that the population size, starting from 40 individuals, becomes 80 individuals after one time unit.

• Doubling time is constant as long as the growth rate remains unchanged.
• It provides a straightforward understanding of growth speed without complex calculations.
• The formula used in exponential growth, such as population doubling, relies on this concept to determine growth parameters.

Understanding doubling time allows us to predict future sizes of populations or investments simply. By knowing how often the doubling occurs, calculations become much simpler when predicting longer time spans.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a mathematical function that helps in solving exponential equations. In this problem, it's used to determine the growth rate \( k \) in the exponential growth formula.

When you have an equation like \( e^{k} = 2 \), taking the natural logarithm of both sides allows you to solve for \( k \) easily. This process involves the property that \( \ln(e^{x}) = x \), which means that applying the natural logarithm and exponential functions essentially reverse each other.

• The natural logarithm is necessary for solving exponential equations involving \( e \), the base of the natural logarithm.
• This concept simplifies the exponential equations allowing for easy calculation of unknowns such as \( k \).
• \( \ln(2) \) specifically gives the rate at which a unit grows to double its size.

You use natural logarithms to change the format of an equation making it linear, which greatly simplifies solving for variables. Thus, they are a powerful tool in exponential growth analysis.

Growth Rate Calculation

The growth rate, denoted as \( k \), quantifies how quickly a population or other quantity grows over time. In the exponential growth equation, it's determined using the doubling condition.

In the exercise, the growth rate is calculated through recognizing that doubling the initial population is key to finding \( k \). Given the exponential equation \( P(t) = P_0 \, e^{kt} \) and knowing that \( P(1) = 80 \), you can deduce that \( e^{k} = 2 \) because the population doubles.

• This calculation aligns with the doubling time, which clarifies the process.
• You use \( k = \ln(2) \) derived from \( e^k = 2 \) to express growth rate through logarithmic transformation.
• Knowing \( k \) helps forecast exponential growth powerfully and predictably.

Understanding the calculation of growth rate allows you to predict how quickly a population will grow, giving you insights into future sizes over any time span. This is crucial for planning or understanding growth patterns in various real-world scenarios.