Problem 53 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

GROWTH OF BACTERIA The following data were compiled by a researcher during the first 10 minutes of an experiment designed to study the growth of bacteria: \begin{tabular}{l|c|c} Number of minutes & 0 & 10 \\ \hline Number of bacteria & 5,000 & 8,000 \end{tabular} Assuming that the number of bacteria grows exponentially, how many bacteria will be present after 30 minutes?

Step-by-Step Solution

Answer

After 30 minutes, there will be approximately 20,995 bacteria.

Step 1: Understand Exponential Growth

Exponential growth can be modeled by the equation Where: - N(t) is the number of bacteria at time t - N_0 is the initial number of bacteria - k is the growth rate - t is the time in minutes

Step 2: Set Up the Initial Conditions

Use the given data to set up the initial conditions: - At t=0, N(0) = 5,000 - At t=10, N(10) = 8,000

Step 3: Determine the Growth Rate

Solve for k using the data points provided. Start by substituting the known values into the exponential growth equation: 8,000 = 5,000 * e^(10*k)To isolate k, follow these steps: 1. Divide both sides by 5,000: 1.6 = e^(10*k) 2. Take the natural logarithm (ln) of both sides to solve for the exponent: ln(1.6) = 10*k 3. Solve for k by dividing both sides by 10: k = ln(1.6)/10 Using a calculator, k= 0.0478

Step 4: Calculate the Number of Bacteria at t=30

Use the determined growth rate to find the number of bacteria at 30 minutes: N(30) = 5,000 * e^(30*0.0478) Evaluate the exponent: 30*0.0478 ≈ 1.434 Use a calculator to find e^1.434 ≈ 4.199 Multiply by the initial number of bacteria: N(30) ≈ 5,000 * 4.199 ≈ 20,995

Key Concepts

Bacterial Growth

Bacteria are fascinating microorganisms that can reproduce quickly under favorable conditions. Bacterial growth typically follows an exponential pattern, meaning the population increases rapidly over time. In this exercise, we are given data observed over 10 minutes to determine how many bacteria will be present after 30 minutes.
When thinking about bacterial growth, it's crucial to understand the concept of exponential growth. Unlike linear growth, where the increase is constant, exponential growth accelerates over time. This is because each generation of bacteria divides and multiplies, increasing the population exponentially. For example, if one bacterium divides into two, those two can then divide into four, and so on. Understanding this, we can better model and predict bacterial population changes over time.

Growth Rate Calculation

The growth rate (k) is a key factor in modeling exponential growth. It tells us how fast the bacteria are multiplying. In our exercise, to find the growth rate, we use the exponential growth formula:
\[N(t) = N_0 \cdot e^{kt}\]
\(N(t)\) is the number of bacteria at time \(t\), \(N_0\) is the initial number of bacteria, and \(e\) is the base of the natural logarithm. Given the data points (t=0, N=5,000) and (t=10, N=8,000), our task is to solve for \(k\).
Substituting the known values into the equation:
\[8,000 = 5,000 \cdot e^{10 \cdot k}\]
We divide both sides by 5,000:
\[1.6 = e^{10k}\]
To isolate \(k\), we take the natural logarithm of both sides:
\[\ln(1.6) = 10k\]
Finally, we solve for \(k\):
\[k = \frac{\ln(1.6)}{10} \approx 0.0478\]

Exponential Functions

Exponential functions are powerful tools for modeling growth processes like bacterial reproduction. In this exercise, we use the exponential growth function to predict the number of bacteria after 30 minutes. Having already calculated the growth rate \(k = 0.0478\), we can apply it to our initial equation to find the number of bacteria at \(t = 30\) minutes.
Using the formula:
\[N(30) = 5,000 \cdot e^{30 \cdot 0.0478}\]
We first calculate the exponent:
\[30 \cdot 0.0478 \approx 1.434\]
Next, we find \(e^{1.434}\) using a calculator:
\[e^{1.434} \approx 4.199\]
Multiplying by the initial number of bacteria, we get:
\[N(30) \approx 5,000 \cdot 4.199 \approx 20,995\]
Therefore, after 30 minutes, there will be approximately 20,995 bacteria. This calculation demonstrates the rapid increase characteristic of exponential growth, emphasizing the importance of understanding exponential functions in biological contexts.