Problem 10 - Algebra and Trigonometry

Question

The count in a culture of bacteria was 400 after 2 hours and 25,600 after 6 hours. (a) What is the relative rate of growth of the bacteria population? Express your answer as a percentage. (b) What was the initial size of the culture? (c) Find a function that models the number of bacteria \(n(t)\) after \(t\) hours. (d) Find the number of bacteria after 4.5 hours. (e) When will the number of bacteria be 50,000?

Step-by-Step Solution

Answer

(a) 103.97% growth rate, (b) initial size 25, (c) model: \( n(t) = 25e^{1.0397t} \), (d) 3,200 bacteria, (e) 7.53 hours.

Step 1: Model the Growth with an Exponential Function

Assume the growth of the bacteria population follows an exponential model, which can be expressed as \( n(t) = n_0 e^{rt} \), where \( n(t) \) is the population at time \( t \), \( n_0 \) is the initial population, \( r \) is the relative growth rate, and \( t \) is the time in hours.

Step 2: Set Up Equations Using Given Data

We know from the problem that \( n(2) = 400 \) and \( n(6) = 25,600 \). Substituting these into our model gives two equations: \( n_0 e^{2r} = 400 \) and \( n_0 e^{6r} = 25,600 \).

Step 3: Solve for the Growth Rate

Dividing the second equation by the first to eliminate \( n_0 \), we get \( \frac{25,600}{400} = e^{4r} \). Simplifying gives \( 64 = e^{4r} \). Taking the natural logarithm of both sides, we have \( 4r = \ln(64) \). So, \( r = \frac{\ln(64)}{4} \approx 1.0397 \). Convert this to a percentage for the growth rate: \( r \approx 103.97\% \).

Step 4: Find the Initial Population

Substituting the value of \( r \) back into the first equation: \( n_0 e^{2 \times 1.0397} = 400 \). Solving for \( n_0 \), we get \( n_0 = \frac{400}{e^{2 \times 1.0397}} \approx 25 \).

Step 5: Write the Exponential Model

Now that we have \( n_0 = 25 \) and \( r = 1.0397 \), we can model the population as \( n(t) = 25e^{1.0397t} \).

Step 6: Calculate the Population at 4.5 Hours

Substituting \( t = 4.5 \) into the model, \( n(4.5) = 25e^{1.0397 \times 4.5} \). Calculating this gives \( n(4.5) \approx 3200 \).

Step 7: Determine When Population Reaches 50,000

Using the model equation \( n(t) = 50,000 \) to solve for \( t \), we set \( 25e^{1.0397t} = 50,000 \). Solving this gives \( e^{1.0397t} = \frac{50,000}{25} \) and \( 1.0397t = \ln(2000) \). Therefore, \( t = \frac{\ln(2000)}{1.0397} \approx 7.53 \) hours.

Key Concepts

Bacteria Population

Bacteria populations are fascinating examples of exponential growth in action. Imagine a single bacterium that divides and multiplies over time, creating a significantly larger population. This type of growth is common in environments where resources like nutrients, temperature, and space are abundant.
Bacteria typically reproduce by binary fission, where one cell splits into two. This quick multiplication means their population size can skyrocket in a short amount of time. In mathematical models, like in our original exercise, bacteria population over time can be described using exponential functions. This helps us predict future population sizes and understand patterns of bacterial growth.
Understanding how bacteria multiply is crucial not only in microbiology but also in medical and environmental fields. It aids in developing strategies to control harmful bacteria or to harness beneficial bacteria for purposes like waste decomposition or fermentative food production.

Growth Rate

The growth rate of a bacteria population indicates how quickly the population is increasing over time. It's important because it tells us the pace at which the bacteria is reproducing.
In the exercise, the growth rate is calculated using an exponential function formulation. We found the growth rate by comparing how the population size changes between two different time points. Specifically, we used data from 2 hours and 6 hours.
After solving the equations, we discovered the growth rate to be approximately 1.0397, which means the population grows by around 103.97% per hour. This rapid rate of growth is a common characteristic of bacteria under ideal conditions.

• Always remember that the growth rate depends on environmental factors, like temperature and nutrient availability.
• Knowing the growth rate helps in predicting how large a bacterial population will become in the future.

Initial Population

The initial population is simply the count of bacteria at the very beginning of the observation or experiment. In any model predicting population growth, knowing where you start is just as crucial as understanding how fast you're growing.
In our example, to find the initial population ( _0 z) of bacteria, we used the provided data points and our calculated growth rate. By rearranging the exponential function model, we determined that the initial count was approximately 25 bacteria.
This number may seem small, but in the context of exponential growth, it provides a crucial baseline for understanding how dramatically the population increases over time. Always keep in mind:

• An accurate initial population ensures our predictions and calculations are reliable.
• It helps in designing and controlling experiments, especially in scientific research.

Exponential Function

Exponential functions are mathematical expressions used to model situations where a quantity grows steadily over time. They're particularly useful in predicting the growth of bacteria populations due to the rapid and continuous nature of bacterial reproduction.
The general form of an exponential function is given by (t = n_0 \times \text{e}^{rt})\, where \(n(t)\) represents the population at time \(t\), \(n_0\) indicates the initial population, \(e\) is the base of the natural logarithm, and \(r\) is the growth rate.
In our exercise, this formula models how a bacteria culture grows from a modest initial size to a considerably larger one. The exponential model reveals that as time progresses, the population can increase exponentially, especially if the conditions remain suitable for growth.

• Exponential functions are not limited to bacteria but are expansive in fields like finance, physics, and demography.
• Learning to use these functions aids in understanding various real-world phenomena marked by rapid change.