Problem 14 - Precalculus Mathematics for Calculus

Question

These exercises use the population growth model. 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) 103.972% growth rate (b) Initial size 100 (c) Model: \(n(t) = 100e^{1.03972t}\) (d) 10746.7 bacteria after 4.5 hours (e) 5.975 hours for 50,000 bacteria

Step 1: Understand the Exponential Growth Formula

The formula for exponential growth is given by:\[ n(t) = n_0 e^{rt} \]where \( n(t) \) is the population at time \( t \), \( n_0 \) is the initial population size, \( r \) is the growth rate, and \( t \) is time.

Step 2: Relate Two Population Values to Find Growth Rate

We know the population at two different times: \( n(2) = 400 \) and \( n(6) = 25600 \). Using the formula,\[ 25600 = 400 e^{4r} \]Solve for \( r \):\( e^{4r} = \frac{25600}{400} = 64 \),\( 4r = \ln 64 \),\( r = \frac{\ln 64}{4} \).

Step 3: Calculate the Growth Rate

Compute \( r \):\[ r = \frac{\ln 64}{4} = \frac{4.15888}{4} = 1.03972 \]Express \( r \) as a percentage: \( r \approx 103.972\% \).

Step 4: Determine Initial Population

Using the exponential growth formula with known \( n(2) = 400 \),\[ 400 = n_0 e^{2r} \]Using \( r \approx 1.03972 \),\[ 400 = n_0 \times e^{2 \times 1.03972} \]Calculate \( n_0 \):\[ n_0 = \frac{400}{e^{2.07944}} = 100 \].

Step 5: Formulate the Exponential Model

Using the values \( n_0 = 100 \) and \( r = 1.03972 \):\[ n(t) = 100 e^{1.03972t} \].

Step 6: Calculate Population After 4.5 Hours

Using the model \( n(t) = 100 e^{1.03972t} \) with \( t = 4.5 \):\[ n(4.5) = 100 e^{1.03972 \times 4.5} \]\[ n(4.5) = 100 \times e^{4.67874} \approx 100 \times 107.467 \approx 10746.7 \].

Step 7: Determine Time for Population to Reach 50,000

Set the model equal to 50000:\[ 50000 = 100 e^{1.03972t} \]Solve for \( t \):\[ 500 = e^{1.03972t} \],\[ t = \frac{\ln 500}{1.03972} \approx \frac{6.21461}{1.03972} \approx 5.975 \] hours.

Key Concepts

Population Growth Model

The population growth model plays a significant role in understanding how populations increase over time. In many scenarios, especially in biology and ecology, populations such as bacteria grow in a way that can be described mathematically.
The basic formula for such growth is the exponential growth equation. This model assumes that the population growth rate is proportional to the current population.

• The model is expressed as: \[ n(t) = n_0 e^{rt} \]
• Where \( n(t) \) is the population at time \( t \)
• \( n_0 \) is the initial population size
• \( r \) is a constant representing the growth rate
• \( e \) is the base of the natural logarithm, approximately equal to 2.71828

This mathematical representation helps in predicting future population sizes and is critical in fields such as microbiology, where understanding the behavior of bacteria over time is vital.

Bacteria Population

A bacteria population is a perfect example of exponential growth in nature. Bacteria reproduce by binary fission, which means one bacterium splits into two, leading to rapid population increase under favorable conditions.
In the context of the given exercise, we consider a culture of bacteria where the initial and changes in population over time can be precisely calculated using the exponential growth model. Here’s a breakdown:

• Given: After 2 hours, the population is 400, and after 6 hours, it becomes 25,600.
• To determine the initial size and behavior over time, use the exponential function model.
• This aids in knowing how many bacterial cells were present initially and predicting population changes over other time periods.

Such calculations are essential for understanding microbial cultures in laboratory settings, ensuring proper handling and experimentation.

Growth Rate

In understanding exponential growth, the growth rate, denoted by \( r \), is crucial. It reflects how quickly the population increases. Often expressed as a percentage, it tells you the rate of increase per unit time.
In our bacterial population problem:

• The growth rate \( r \) is computed based on given population values at different times.
• You can calculate it by rearranging the exponential growth equation and solving for \( r \).
• Here, \( r \) was determined using the known populations \( n(2) = 400 \) and \( n(6) = 25600 \).
• The resultant growth rate was found to be approximately 103.972%.

The growth rate provides insights into how aggressively the bacteria multiply, and this information can be vital for processes like brewing, fermentation, or infection control.

Exponential Function

The exponential function is a mathematical function that models situations where growth or decay occurs at a constant rate. In the context of the population growth model, it is used to describe how quantities like populations change over time exponentially.

• The basic form of an exponential function is \( f(t) = a e^{bt} \), where \( a \) is the initial amount and \( b \) is a constant that determines the rate of growth or decay.
• In our example, \( n(t) = 100 e^{1.03972t} \), this exponential function accurately describes the bacteria population over time.
• It allows predicting how population sizes will change at any point.

Applying this function, you can determine both the past and future states of populations in various scenarios, making it an indispensable tool in fields ranging from biology to economics.