Problem 8 - Algebra and Trigonometry

Question

A culture contains 1500 bacteria initially and doubles every 30 min. (a) Find a function that models the number of bacteria \(n(t)\) after \(t\) minutes. (b) Find the number of bacteria after 2 hours. (c) After how many minutes will the culture contain 4000 bacteria?

Step-by-Step Solution

Answer

(a) \( n(t) = 1500 \times 2^{t/30} \) (b) The number of bacteria after 2 hours is 24000. (c) It will take approximately 47.53 minutes to reach 4000 bacteria.

Step 1: Understanding the Exponential Growth Formula

We know that the population of bacteria grows exponentially. For exponential growth, the population can be modeled as \( n(t) = n_0 \times 2^{t/T} \), where \( n_0 \) is the initial number of bacteria, and \( T \) is the doubling time in minutes.

Step 2: Apply Initial Values to the Model

The initial number of bacteria \( n_0 = 1500 \), and the doubling time \( T = 30 \) minutes. Thus, the function modeling the number of bacteria after \( t \) minutes is \( n(t) = 1500 \times 2^{t/30} \).

Step 3: Calculate the Number of Bacteria After 2 Hours

To find the number of bacteria after 2 hours, we convert hours to minutes: 2 hours = 120 minutes. Substitute \( t = 120 \) into the model: \( n(120) = 1500 \times 2^{120/30} = 1500 \times 2^4 = 1500 \times 16 = 24000 \).

Step 4: Solve for Time Until Population Reaches 4000 Bacteria

To find the time at which the culture reaches 4000 bacteria, set \( n(t) = 4000 \). Thus, \( 1500 \times 2^{t/30} = 4000 \). Divide both sides by 1500: \( 2^{t/30} = \frac{4000}{1500} = \frac{8}{3} \).

Step 5: Use Logarithms to Solve for Time

Take the logarithm of both sides: \( \log(2^{t/30}) = \log \left(\frac{8}{3}\right) \). This simplifies to \( \frac{t}{30} \log(2) = \log \left(\frac{8}{3}\right) \). Solve for \( t \): \( t = 30 \times \frac{\log(8/3)}{\log(2)} \approx 47.53 \) minutes.

Key Concepts

Bacteria Population

Bacteria populations are fascinating examples of exponential growth in nature. When you have a starting number of bacteria, like 1500 in this scenario, they grow by continually multiplying over time. Understanding the growth pattern of bacteria helps in various fields, from medicine to environmental science.
Bacteria reproduce by dividing, which means that each bacterium splits into two, essentially doubling the population. When examining these populations, scientists often refer to initial populations, denoted as \(n_0\), as the number from which growth begins. In our case, with 1500 bacteria initially, this value forms the base of further calculations for population growth.

Doubling Time

Doubling time is a key feature of exponential growth, especially with bacteria. It is the time it takes for a population to double in size. For the bacteria culture in the exercise, doubling time is given as 30 minutes.
This means every half hour, the number of bacteria will become twice as large. This predictable pattern allows us to model population growth easily. You can think of doubling time as a sort of biological clock that ticks every 30 minutes in this example, leading to rapid increases in number. Understanding doubling time is crucial for predicting how quickly populations can grow.

Exponential Function

An exponential function is a mathematically powerful way to express how a quantity grows quickly over time. In this case, the number of bacteria over time is represented with the formula \(n(t) = n_0 \times 2^{t/T}\).
Here, \(t\) represents elapsed time, \(T\) is the doubling time, and the base 2 indicates the doubling nature of growth. With an initial population of 1500 and a doubling time of 30 minutes, our bacteria growth can be neatly expressed through this function.
Exponential functions are perfect for modeling processes that involve growth or decay, such as populations or radioactive decay, because they account for continuous growth rates.

Logarithms

Logarithms are the mathematical tool we use to solve equations involving exponential growth when calculating time. In our exercise, they are used to find how long it will take the bacteria population to reach 4000.
We set up the equation \(1500 \times 2^{t/30} = 4000\) and isolate the exponential term. Then, we use logarithms to solve - taking the log of both sides helps us deal with the exponents, leading to manageable calculations.
Logarithms convert multiplication into addition, making it easier to find the unknown \(t\). This approach is incredibly useful in biology and other sciences whenever exponential growth or decay needs to be unraveled into simpler calculations.