Problem 40 - Calculus Early Transcendentals

Question

Find a formula for the tripling time of an exponential growth model.

Step-by-Step Solution

Answer

The tripling time is \( T = \frac{\ln(3)}{r} \).

Step 1: Understand the Exponential Growth Model

In an exponential growth model, the quantity increases according to the formula \( P(t) = P_0 \cdot e^{rt} \), where \( P(t) \) is the amount at time \( t \), \( P_0 \) is the initial amount, \( r \) is the growth rate, and \( e \) is the base of the natural logarithm.

Step 2: Set Up the Equation for Tripling Time

We want to find the time \( T \) when the initial amount \( P_0 \) triples, which means \( P(T) = 3P_0 \). Substituting this into the exponential growth model gives \( 3P_0 = P_0 \cdot e^{rT} \).

Step 3: Simplify the Equation

Divide both sides of the equation by \( P_0 \) to isolate the exponential term: \( 3 = e^{rT} \).

Step 4: Solve for the Tripling Time

Taking the natural logarithm of both sides will help solve for \( rT \): \( \ln(3) = rT \).

Step 5: Solve for \( T \)

Divide both sides by \( r \) to find \( T \): \( T = \frac{\ln(3)}{r} \). This is the formula for the tripling time in an exponential growth model.

Key Concepts

Tripling Time

Tripling time refers to the period it takes for a quantity to grow to three times its original size under exponential growth conditions. Understanding how to calculate this is essential for analyzing situations such as population growth, financial investments, or any phenomena that can be described with an exponential growth model.

The concept of tripling time is directly connected to the growth rate of the system. In simpler terms, tripling time helps you determine how swiftly the process you are observing actually triples. The formula derived for tripling time is given by:

\[ T = \frac{\ln(3)}{r} \]

where \( T \) is the tripling time, \( \ln(3) \) refers to the natural logarithm of 3, and \( r \) is the growth rate. This equation encapsulates how the initial amount impacts the necessary time for tripling is neutralized, ensuring the calculation focuses solely on the intrinsic growth characteristic of the system.

Exponential Growth Model

Exponential growth model describes a situation where a quantity grows at a rate proportional to its current size. This self-multiplying behavior can be expressed mathematically as:

\[ P(t) = P_0 \cdot e^{rt} \]

Here, \( P(t) \) represents the quantity at time \( t \), \( P_0 \) is the original amount, \( r \) indicates the growth rate, and \( e \) is approximately equal to 2.71828, the base of the natural logarithm. This model is useful whenever growth accelerates over time, creating a curve that rises ever more steeply.

Using exponential models allows us to make predictions about future quantities based on the current state and growth rate. This appears in many fields such as biology for population modeling, economics in estimating compound interest, and many natural processes where growth rate increases over time.

Natural Logarithm

The natural logarithm is a mathematical function denoted as \( \ln(x) \) which solves where \( e^y = x \). In other words, it tells you what power you would need to raise \( e \) to in order to obtain \( x \). The constant \( e \), roughly 2.71828, presents naturally across exponential growth and decay scenarios.

Understanding logarithms helps in simplifying expressions and equations, particularly when dealing with exponential growth models. For example, the natural logarithm is crucial in determining the tripling time. By transforming the equation \( 3 = e^{rT} \) into \( \ln(3) = rT \), you can solve for time \( T \) with ease. Keeping in mind the properties of logarithms can simplify complex mathematical problems, opening the path to more straightforward calculations.