Problem 34 - Calculus

Question

A culture of bacteria originally numbers \(500 .\) After 2 hours there are 1500 bacteria in the culture. Assuming exponential growth, how many are there after 6 hours?

Step-by-Step Solution

Answer

After 6 hours, there are 13,500 bacteria.

Step 1: Understand Exponential Growth Formula

Exponential growth is defined by the formula \( N(t) = N_0 e^{rt} \), where \( N(t) \) is the number of bacteria at time \( t \), \( N_0 \) is the initial number of bacteria, \( r \) is the growth rate, and \( e \) is the base of the natural logarithm.

Step 2: Identify Known Values

Given that \( N_0 = 500 \) and \( N(2) = 1500 \), we can set up the equation \( 1500 = 500 \, e^{2r} \) to find the growth rate \( r \).

Step 3: Solve for Growth Rate \( r \)

Rearrange the equation \( 1500 = 500 \, e^{2r} \) to \( e^{2r} = 3 \). Taking the natural logarithm of both sides gives \( 2r = \ln(3) \). Thus, \( r = \frac{\ln(3)}{2} \).

Step 4: Calculate Number of Bacteria after 6 Hours

Use \( r \) to find the number of bacteria after 6 hours. Substitute into the formula: \( N(6) = 500 \, e^{6r} = 500 \, e^{6 \left(\frac{\ln(3)}{2}\right)} \). Simplify the exponent and calculate \( N(6) \).

Step 5: Simplify and Solve the Final Expression

Since \( 6 \times \frac{\ln(3)}{2} = 3 \ln(3) \), \( N(6) = 500 \, (e^{\ln(3)})^3 = 500 \, 3^3 = 500 \times 27 = 13500 \).

Key Concepts

Bacteria Growth

Bacteria growth is a fascinating process that often exhibits exponential patterns. In this context, we have a situation where a culture of bacteria grows from 500 to 1500 in just 2 hours. This significant increase can be attributed to their exponential growth nature.

Exponential growth means that the number of bacteria increases by a consistent percentage over equal time intervals. This translates into the population growing faster as time progresses because new bacteria also reproduce. Often, you will see this modeled with the formula:

• \( N(t) = N_0 e^{rt} \)

In this equation, \( N(t) \) represents the number of bacteria at time \( t \), \( N_0 \) is the initial number, \( e \) is the base of the natural logarithm, and \( r \) is the growth rate.

This model is commonly used due to its accuracy in predicting growth under ideal conditions, meaning no resource limitations or environmental changes.

Growth Rate Calculation

Calculating the growth rate is a crucial step in understanding exponential trends. In our bacteria example, we start by identifying known values: initially 500 bacteria growing to 1500 in 2 hours. We use the equation \( 1500 = 500 \, e^{2r} \) to find the growth rate \( r \).

Here's a simple way to approach it:

• First, simplify the equation: \( e^{2r} = 3 \).
• To solve for \( r \), take the natural logarithm on both sides: \( 2r = \ln(3) \).
• Now, isolate \( r \) by dividing by 2: \( r = \frac{\ln(3)}{2} \).

With this growth rate, we can predict how many bacteria will be there after a different period, say, 6 hours.

Natural Logarithm

Natural logarithms are particularly useful in exponential growth problems. They help in simplifying calculations and solving for variables like growth rate. The natural logarithm, denoted as \( \ln \), is the logarithm to the base \( e \), where \( e \approx 2.71828 \).

The utility of natural logarithms lies in their ability to linearize exponential equations. For instance, in our solution, to solve \( e^{2r} = 3 \), we take the natural logarithm of both sides to get \( 2r = \ln(3) \). This step simplifies the exponential equation into a linear one, making it much easier to solve.

Using natural logarithms, we turn a multiplicative growth model into an additive form, thus facilitating straightforward calculations for values like the growth rate. This application is essential knowledge for tackling many scientific equations and real-world problems involving growth and decay.