Problem 1 - Algebra and Trigonometry

Question

Bacteria Culture A certain culture of the bacterium Streptococcus \(A\) initially has 10 bacteria and is observed to double every 1.5 hours. (a) Find an exponential model \(n(t)=n_{0} 2^{t / 2}\) for the number of bacteria in the culture after \(t\) hours. (b) Estimate the number of bacteria after 35 hours. (c) When will the bacteria count reach \(10,000 ?\)

Step-by-Step Solution

Answer

(a) Model: \(n(t) = 10 \times 2^{t/1.5}\). (b) After 35 hours: ~100,000,000 bacteria. (c) 10,000 bacteria in ~14.95 hours.

Step 1: Identify Initial Values

The problem states that the initial number of bacteria is 10, so we have \( n_0 = 10 \). The bacteria double every 1.5 hours, which can be incorporated into the model later.

Step 2: Formulate the Exponential Model

Since the bacteria double every 1.5 hours, we express the growth as an exponent based on the doubling period. The formula given is \( n(t) = n_0 \times 2^{t/1.5} \). Substituting the initial value \( n_0 = 10 \) into the equation, the model becomes \( n(t) = 10 \times 2^{t/1.5} \).

Step 3: Estimate Bacteria Count After 35 Hours

To find the number of bacteria after 35 hours, substitute \( t = 35 \) into the model: \[ n(35) = 10 \times 2^{35/1.5} \]Calculate the expression: \[ n(35) \approx 10 \times 2^{23.3333} \approx 10 \times 10^7.02 \approx 10^8 \]This gives approximately 100,000,000 bacteria.

Step 4: Determine Time to Reach 10,000 Bacteria

Set the equation \( n(t) = 10,000 \) to find \( t \): \[ 10,000 = 10 \times 2^{t/1.5} \]Divide by 10 to get \[ 1,000 = 2^{t/1.5} \]Take the logarithm base 2 to solve for \( t \): \[ \log_2(1,000) = \frac{t}{1.5} \]Calculate \( \log_2(1,000) \approx 9.96578 \), then solve for \( t \): \[ t = 9.96578 \times 1.5 \approx 14.94867 \]This rounds to approximately 14.95 hours.

Key Concepts

Bacteria Culture

Bacteria cultures are fascinating examples of exponential growth in nature. In this exercise, we examine a culture of Streptococcus bacteria. Initially, this culture contains 10 bacteria. This is referred to as the initial population size, denoted as \( n_0 \). Bacteria, like many microorganisms, reproduce rapidly under ideal conditions. This makes them excellent subjects for studying exponential growth. As the bacteria reproduce, they double in number, meaning the population size increases by a factor of two during each period of time known as the doubling time. Understanding these basic concepts helps in mathematically modeling and predicting how bacteria populations grow over time. By knowing the initial population and the rate at which the population doubles, one can calculate the size of the population at any future time point. This is crucial for laboratories dealing with bacteria cultures on a regular basis.

Doubling Time

Doubling time is a fundamental concept when analyzing exponential growth, particularly in biological systems like bacteria cultures. It is the period of time it takes for a population to double in size. In our Streptococcus culture example, the doubling time is 1.5 hours. Understanding doubling time enables us to create mathematical models that accurately predict population growth. It is particularly useful when estimating how fast a population grows and when it will reach a certain size. This can be illustrated using the exponential model \( n(t) = n_0 \times 2^{t/1.5} \), where \( t \) is time in hours, and the factor \( 2^{t/1.5} \) represents the exponential growth related to the doubling period. Doubling time is important not just in the context of bacteria, but it is widely applied in other areas such as finance (for investments), physics, and environmental studies, allowing for a consistent method of calculating growth across different domains.

Mathematical Modeling

Mathematical modeling helps provide a clear picture of how systems such as bacteria cultures behave over time. For the Streptococcus culture, a simple yet powerful exponential model \( n(t) = 10 \times 2^{t/1.5} \) describes the bacterial growth. This model is based on the concept of exponential growth, where a constant factor multiplies the original amount. Here, each 1.5 hours of doubling time is incorporated into the exponent \( t/1.5 \). Creating such models requires understanding initial conditions and growth rates. After establishing the model, it can be used to estimate future behavior, such as:

• Estimating the number of bacteria after a given time, as demonstrated with 35 hours resulting in about 100,000,000 bacteria.
• Determining when a certain population size will be reached, like 10,000 bacteria occurring approximately at 14.95 hours.

Mathematical modeling is a powerful tool that, once mastered, allows for the analysis and prediction of complex real-world phenomena, not limited to microbiology, but also in economics, epidemiology, and engineering.