Problem 23 - Precalculus

Question

Under optimal conditions, the growth of a certain strain of \(E\). Coli is modeled by the Law of Uninhibited Growth \(N(t)=N_{0} e^{k t}\) where \(N_{0}\) is the initial number of bacteria and \(t\) is the elapsed time, measured in minutes. From numerous experiments, it has been determined that the doubling time of this organism is 20 minutes. Suppose 1000 bacteria are present initially. (a) Find the growth constant \(k\). Round your answer to four decimal places. (b) Find a function which gives the number of bacteria \(N(t)\) after \(t\) minutes. (c) How long until there are 9000 bacteria? Round your answer to the nearest minute.

Step-by-Step Solution

Answer

(a) The growth constant is approximately 0.0347. (b) The function is \(N(t) = 1000 \cdot e^{0.0347t}\). (c) It takes approximately 67 minutes to reach 9000 bacteria.

Step 1: Understanding the doubling time

Doubling time is the time it takes for a quantity to double in size or value. For this bacteria, it has a doubling time of 20 minutes. This means in 20 minutes, the initial count of bacteria will double.

Step 2: Finding the growth constant k

The formula for doubling time is given by the equation \( 2 = e^{k imes 20} \). Solve for \( k \) by taking the natural logarithm of both sides:\[\ln(2) = k imes 20 \k = \frac{\ln(2)}{20} \k \approx 0.0347\]

Step 3: Writing the function N(t)

Use the growth constant \(k\) found in Step 2 and the initial bacteria count \(N_0 = 1000\) in the exponential growth formula:\[ N(t) = 1000 \cdot e^{0.0347t} \]

Step 4: Solving for the time when N = 9000

Set \(N(t) = 9000\) and solve for \(t\):\[9000 = 1000 \cdot e^{0.0347t} \9 = e^{0.0347t} \\ln(9) = 0.0347t \t = \frac{\ln(9)}{0.0347} \t \approx 66.8\]Round this value to the nearest minute: \(t \approx 67\) minutes.

Key Concepts

Doubling Time

Doubling time is a term often used in the context of exponential growth, especially when studying populations of living organisms, such as bacteria. Doubling time is the period of time required for a quantity to double in size. It applies to anything growing at a constant proportion, not just bacteria. Here’s how it is determined:

• Identify the initial amount.
• Measure how long it takes for this amount to become twice as large.

In the example of the bacterial growth problem, the strain of bacteria has a doubling time of 20 minutes. Understanding doubling time allows you to see how quickly something grows. For example, if you start with 1000 bacteria, in 20 minutes, you’d have 2000, then 4000 after another 20 minutes, and so on. This concept of doubling is a classic hallmark of exponential growth.

Growth Constant

The growth constant in exponential growth models describes how quickly something is growing. It is represented by the variable \(k\) in the mathematical model \(N(t) = N_0 e^{kt}\). To find the growth constant, you can use the formula for doubling time: \[2 = e^{k \times \, ext{Doubling Time}}\] By solving for \(k\) through logarithms:

• Take the natural log of both sides: \(\ln(2) = k \times \, ext{Doubling Time}\)
• Rearrange to solve for \(k\): \(k = \frac{\ln(2)}{\text{Doubling Time}}\)

For our bacterial example, it is calculated as:\[k = \frac{\ln(2)}{20} \approx 0.0347\]This constant, \(k\), is crucial because it determines how fast the bacteria population grows over time. A larger \(k\) means faster growth.

Bacterial Growth

Bacterial growth is a fascinating and rapid process, typically modeled by exponential functions due to their capacity for uninhibited expansion under optimal conditions. In microbiology, understanding bacterial growth involves several phases:

• Lag Phase: Adjusting to the environment.
• Exponential Phase: Rapid growth, which we are modeling here.
• Stationary Phase: Growth rates slow as resources become limited.
• Death Phase: Decline due to resource depletion.

For exponential growth, the model \(N(t) = N_0 e^{kt}\) helps predict the population size at any given time \(t\). Starting with an initial population \(N_0 = 1000\), using the growth constant \(k = 0.0347\), the growth function becomes:\[N(t) = 1000 \cdot e^{0.0347t}\]This function provides a simple way to calculate the future population size, such as finding out when the population reaches 9000.

Exponential Functions

Exponential functions are a cornerstone of mathematical modeling in natural sciences, representing processes where the rate of change is proportional to the current value. They are generally expressed as \(f(x) = a \cdot e^{bx}\), where \(e\) is Euler’s number (approximately 2.71828). These functions show how quantities grow rapidly over time.In the context of bacterial growth, an exponential function is used because it perfectly models scenarios where a population doubles regularly. Characteristic features of exponential functions include:

• Rapid Increase: As seen in the example, bacterial counts surge rapidly once doubling time is reached.
• Constant Growth Rate: The growth constant \(k\) dictates how quickly the population increases.
• Continual Doubling: The population doubles at regular intervals, making it predictable and easy to calculate future values.

Exponential functions are versatile tools not only for modeling bacteria but also in fields like finance and ecology, where entities grow multiplicatively over time.