Problem 88 - Algebra and Trigonometry

Question

Bacteria Colony A certain strain of bacteria divides every three hours. If a colony is started with 50 bacteria, then the time \(t\) (in hours) required for the colony to grow to \(N\) bacteria is given by $$ t=3 \frac{\log (N / 50)}{\log 2} $$ Find the time required for the colony to grow to a million bacteria.

Step-by-Step Solution

Answer

The colony will take approximately 42.89 hours to grow to a million bacteria.

Step 1: Understand the Problem

We need to find the time \( t \) required for an initial colony of 50 bacteria to grow to a population of 1,000,000 bacteria using the given formula. The formula we use is \[ t = 3 \frac{\log (N / 50)}{\log 2} \] where \( N = 1,000,000 \).

Step 2: Substitute the Value of N

Put the desired number of bacteria \( N = 1,000,000 \) into the formula. The equation becomes: \[ t = 3 \frac{\log (1,000,000 / 50)}{\log 2} \]

Step 3: Simplify the Fraction Inside the Logarithm

First, simplify the fraction \( \frac{1,000,000}{50} \), which equals 20,000. The equation is now: \[ t = 3 \frac{\log 20,000}{\log 2} \]

Step 4: Calculate the Logarithms

Now, calculate the logarithm values. \( \log_{10}(20,000) \approx 4.3010 \) and \( \log_{10}(2) \approx 0.3010 \). Substitute these into the equation: \[ t = 3 \frac{4.3010}{0.3010} \]

Step 5: Perform Division Inside the Formula

Divide 4.3010 by 0.3010 to get approximately 14.2967. So, the equation now reads: \[ t = 3 \times 14.2967 \]

Step 6: Multiply to Find Time t

Finally, multiply 3 by 14.2967 to find \( t \). \( t \approx 42.89 \).

Step 7: Round the Result

Since time is typically measured in hours, round \( t \) to two decimal places, so \( t \approx 42.89 \) hours.

Key Concepts

Logarithms

Logarithms are fundamental in mathematical calculations, especially when dealing with exponential growth, such as bacteria colonies. A logarithm essentially answers the question: "To what power do we raise the base to get a certain number?" For example, in the equation \( \, \log_{10}(20,000) = 4.3010 \, \, \), it means that 10 raised to the power of 4.3010 equals 20,000. This concept helps simplify multiplicative processes through addition.

• The formula for logarithms is \( \, \, \, b^x = y \, \, \, \) becomes \( \, \, \, x = \, \, \, \, \log_b(y) \, \, \, \).
• They are particularly useful in solving equations where the unknown is an exponent.
• In exponential growth calculations, logarithms allow us to determine how long it takes for a population to reach a certain size.

Understanding logarithms can greatly simplify and solve problems related to growth and decay. They help transform multiplicative relationships (like those found in bacteria growth) into additive ones, providing a manageable pathway to find solutions.

Bacteria Colony Growth

Bacteria multiply rapidly, following a pattern called exponential growth. Exponential growth occurs when the growth rate of a value is proportional to its current size. For bacteria, this means the population size doubles at consistent intervals, known as the doubling time. For instance, in this exercise, the bacteria colony doubles every three hours.

• The initial population can be represented as \( P_0 \), and doubles to \( 2P_0 \), \( 4P_0 \), etc.
• In the case of the exercise, with an initial population of 50, the pattern follows: 50, 100, 200, 400, and so on.
• The growth pattern is modeled by the formula \( N = P_0 \, \, \, \, \times 2^{t/3} \), which describes how the population grows over time \( t \).

By understanding this pattern, we can observe how quickly bacteria can reproduce and reach significant numbers in a short period. It's essential in fields like microbiology and medicine, where predicting bacteria growth can inform treatment and prevention strategies.

Mathematical Modeling

Mathematical modeling is a powerful tool for representing real-world phenomena through mathematical formulas and equations. In the context of bacteria growth, it's used to predict how a colony will expand over time. This process involves understanding initial conditions, like the starting population size, and applying known growth patterns or formulas.

• The model derived for bacteria colony growth here is \( t = 3 \, \, \, \, \frac{\log(N / 50)}{\log 2} \), which calculates the time required to reach a population size \( N \).
• Such models rely on simplifying assumptions, like constant growth rates, to make complex systems comprehensible.
• They facilitate prediction and analysis, allowing us to forecast outcomes and plan accordingly.

The aim of mathematical modeling is not only to predict future states but also to understand underlying dynamics of systems. In our example, we predict the time to reach 1,000,000 bacteria, giving us insight into the rapidity of bacterial multiplication.