Problem 44 - Algebra 2

Question

Biology For Exercises \(44-46,\) use the formula \(H=\left(\begin{array}{c}{1} \\\ {r}\end{array}\right)(\ln P-\ln A) . H\) is the number of hours, \(r\) is the rate of decine, \(P\) is the initial bacteria population, and \(A\) is the reduced bacteria population. A scientist determines that an antibiotic reduces a population of \(20,000\) bacteria to 5000 in 24 hours. Find the rate of decline caused by the antibiotic.

Step-by-Step Solution

Answer

The rate of decline caused by the antibiotic is \(r = \frac{24}{\ln 4}\).

Step 1: Arrange the equation for 'r'

The first step will be to isolate 'r' mentioned in the formula by manipulating the equation. The original formula can be rewritten as: \[ r=\frac{H}{\ln P-\ln A}\]

Step 2: Substitute the given values

Next, strategically substitute the given values into the equation where \(H = 24\), \(P = 20000\) and \(A = 5000\). Hence, \[r = \frac{24}{\ln 20000 - \ln 5000}\]

Step 3: Simplify the Equation

Simplify the equation by using the property of logarithms to combine the terms in the denominator which results in: \[ r = \frac{24}{\ln \left( \frac{20000}{5000} \right)} \] Simplify further to: \[ r = \frac{24}{\ln 4}\]

Key Concepts

Logarithmic Functions in Exponential Decay

Logarithmic functions play a crucial role in understanding exponential decay, especially in scenarios involving population changes, such as bacteria populations. Logarithms are the inverse operations of exponentiation. In simple terms, if you have an equation involving exponentials, logarithms help solve for the variable. For example, if you wanted to figure out how many times you need to raise a number to get a certain result, you'd use a logarithm. This is because logarithmic functions can simplify exponential equations into linear ones, making them much easier to work with.
The formula used in this problem, involving natural logarithms \((\ln)\), helps find the rate of decline in a bacteria population. The logarithmic function makes calculating changes much more straightforward by utilizing the properties of logs, such as condensing and expanding expressions.

Bacteria Population Dynamics

Bacteria populations can grow and decline exponentially. Exponential growth and decay are processes that increase or decrease at a rate proportional to the current amount. In this context, the initial bacteria population \(P\) was found to be about 20,000, and it decreased to about 5,000 over a specific time frame.
Understanding how exponential decay affects populations can help scientists and researchers develop strategies to control bacteria efficiently. By measuring changes in population size over time, as well as understanding and predicting these changes, it's possible to determine effective treatment doses or infection control methods.

• The initial count, \(P\), reflects the population size before any decline.
• The reduced count, \(A\), is the population size after a period of time under certain conditions.

The knowledge of population dynamics is crucial in fields such as biology and medicine.

Understanding Rate of Decline

The rate of decline \(r\) is an essential factor when studying population changes under certain conditions, especially with substances like antibiotics. In general, this rate tells us how fast a population decreases over time. It's calculated by measuring the decrease in a population over a given period, then applying mathematical formulas to understand this change in depth.
In the provided exercise, calculating the rate of decline involved isolating the variable 'r' and then solving it using known values. The given task involved substituting specific numerical values for time \(H\), initial \(P\), and reduced populations \(A\), then simplifying using logarithms: \[ r = \frac{H}{\ln \frac{P}{A}} \]
Understanding this process is crucial for accurately quantifying and predicting how populations adjust in various scenarios, making it invaluable for researchers and engineers across many disciplines.