Problem 396 - Precalculus

Question

The formula for an increasing population is given by \(P(t)=P_{0} e^{r t}\) where \(P_{0}\) is the initial population and \(r> 0\) . Derive a general formula for the time \(t\) t takes for the population to increase by a factor of \(M .\)

Step-by-Step Solution

Answer

The general formula for time \( t \) is \( t = \frac{\ln(M)}{r} \).

Step 1: Express Population Increase by Factor M

To determine the time it takes for the population to increase by a factor of \( M \), we need to express this condition mathematically. We start with the expression for the population \( P(t) \) at time \( t \) and set it equal to the initial population \( P_0 \) multiplied by \( M \). Thus, we have:\[P(t) = M \cdot P_0\]

Step 2: Substitute the Growth Formula

Next, substitute the given formula for population growth into the equation. Replace \( P(t) \) with \( P_0 e^{rt} \) in the equation from the previous step:\[P_0 e^{rt} = M \cdot P_0\]

Step 3: Cancel the Initial Population

Divide both sides of the equation by \( P_0 \) to simplify the equation. Assuming \( P_0 eq 0 \), we get:\[e^{rt} = M\]

Step 4: Isolate the Exponential Component

To isolate \( t \), take the natural logarithm of both sides. This process uses the property that \( \, \ln(e^{x}) = x \, \):\[\ln(e^{rt}) = \ln(M)\]

Step 5: Solve for Time t

The equation \( \ln(e^{rt}) = \ln(M) \) simplifies to \( rt = \ln(M) \). Now, solve for \( t \) by dividing both sides by \( r \):\[t = \frac{\ln(M)}{r}\]

Key Concepts

Population Growth Formula

Understanding the population growth formula is crucial when studying how populations of organisms evolve over time. The formula usually looks like this: \[ P(t) = P_{0} e^{rt} \]where:

• \( P(t) \) is the population at time \( t \).
• \( P_{0} \) represents the initial population at the starting point (\( t = 0 \)).
• \( r \) is the growth rate, which should be greater than 0 for populations that are increasing.
• \( t \) is time.

Exponential growth occurs because the rate of growth is directly proportional to the current size of the population. This means that as the population grows, the growth rate applies to a larger base, thereby increasing the growth rate exponentially over time.
This mathematical model is essential for predicting how populations will change, given a constant rate over a specific period. It's widely used in demographics, economics, and ecology to understand population dynamics and predict future changes.

Natural Logarithm

The concept of the natural logarithm is essential when dealing with exponential functions, particularly in simplifying exponential equations. The natural logarithm, denoted as \( \ln \), is the logarithm to the base \( e \), where \( e \approx 2.71828 \).
This logarithm is a door-opener to understanding and solving exponential-related problems. It has a unique property:

• Understanding \( \ln(e^{x}) = x \). This means that taking the natural logarithm of an exponential function cancels out the exponent’s base \( e \).

When we write \( \ln(e^{rt}) \), the exponential part simplifies to \( rt \).
The natural logarithm helps us isolate variables within exponential equations, making it possible to solve for unknowns, like time (\( t \)) or growth rate (\( r \)), especially in equations derived from population models.

Time Calculation in Exponential Functions

Calculating time in the context of exponential population growth involves some specific steps once we have our exponential growth equation. The central problem often revolves around determining how long it takes for a population to multiply by a factor. Using our previous discussion of natural logarithms, the time \( t \) can be extracted using the following sequences:
Given the equation\[ e^{rt} = M \]we can take the natural logarithm of both sides,
so we obtain\[ \ln(e^{rt}) = \ln(M) \],
which simplifies to \[ rt = \ln(M) \].

• To solve explicitly for \( t \), divide both sides by the growth rate \( r \): \[ t = \frac{\ln(M)}{r} \].

This equation reveals that the time it takes for a population to increase by a factor \( M \) depends on both the growth rate \( r \) and the natural logarithm of the factor \( M \).
Such calculations are common in population studies, allowing researchers to predict when a population might reach a particular size given its current growth trend.