Problem 64 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

BACTERIAL GROWTH The number of bacteria in a certain culture grows exponentially. If 5,000 bacteria were initially present and 8,000 were present 10 minutes later, how long will it take for the number of bacteria to double?

Step-by-Step Solution

Answer

It will take approximately 14.5 minutes for the bacteria to double.

Step 1 - Define the exponential growth formula

The exponential growth of bacteria can be modeled by the equation \[ N(t) = N_0 e^{kt} \] where \( N(t) \) is the number of bacteria at time \( t \), \( N_0 \) is the initial number of bacteria, and \( k \) is the growth rate constant.

Step 2 - Identify the given values

From the problem, we know that initially, \( N_0 = 5000 \). After 10 minutes, \( N(10) = 8000 \).

Step 3 - Plug in the known values to find \( k \)

Use the equation \( N(t) = N_0 e^{kt} \) with \( t = 10 \) minutes: \[ 8000 = 5000 e^{10k} \] Solve for \( k \): \[ \frac{8000}{5000} = e^{10k} \] \[ 1.6 = e^{10k} \] Take the natural logarithm of both sides: \[ \text{ln}(1.6) = 10k \] \[ k = \frac{\text{ln}(1.6)}{10} \]

Step 4 - Calculate \( k \)

Find the value of \( k \): \[ k = \frac{\text{ln}(1.6)}{10} \approx 0.0478 \]

Step 5 - Determine the time for the bacteria to double

The bacteria double when \( N(t) = 2N_0 \). Thus, we need to find \( t \) when \[ 2N_0 = N_0 e^{kt} \] Divide both sides by \( N_0 \): \[ 2 = e^{kt} \] Take the natural logarithm of both sides: \[ \text{ln}(2) = kt \] Solve for \( t \): \[ t = \frac{\text{ln}(2)}{k} \]

Step 6 - Calculate the doubling time

Substitute \( k \) into the equation: \[ t = \frac{\text{ln}(2)}{0.0478} \approx 14.5 \text{ minutes} \]

Key Concepts

Bacterial Growth

Bacterial growth refers to the increase in the number of bacteria in a population. This phenomenon is commonly seen in bacterial cultures in a lab or natural environment.
Bacteria reproduce through a process called binary fission, where a single bacterium divides into two identical daughter cells. Under optimal conditions, this process leads to exponential growth, meaning the population doubles over regular intervals.
Understanding bacterial growth is crucial in fields like medicine, where controlling harmful bacteria is essential, and in biotechnology, where beneficial bacteria are cultivated.

Exponential Function

The exponential function is a mathematical concept representing growth or decay processes, often expressed as \(\text{N(t) = N_0 e^{kt}}\).
Here, \(N(t)\) represents the number of entities at time \(t\), \(N_0\) is the initial quantity, \(e\) is the base of the natural logarithm (approximately 2.718), and \(k\) is the growth rate constant.
For bacterial growth, the exponential function models how the population increases over time. The nature of exponential functions means that small changes in time result in significant changes in population size, especially over longer periods.

Growth Rate Constant

The growth rate constant \( k \) is a crucial parameter in the exponential growth model. It quantifies how fast the population grows.
In our example, we calculated \( k \) using the known values of initial and later population sizes over a time period. To determine \( k \), we used the equation \( \text{ln}(1.6) = 10k \).
After solving, we found \( k \approx 0.0478 \). This constant helps predict future population sizes and calculate other important metrics like doubling time.

Doubling Time

Doubling time is the period it takes for a population to double in size. For bacterial cultures, it's essential to understand how quickly the population can expand.
We determined the doubling time using the equation \(\text{t} = \frac{\text{ln}(2)}{k}\).
Substituting \(k \approx 0.0478\), we found the doubling time to be approximately 14.5 minutes. This rapid increase exemplifies exponential growth, where small time increments lead to drastically larger populations.