Problem 45 - 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 laboratory assistant tests an antibiotic that causes a rate of decline of 0.14 . How long should it take for a population of 8000 bacteria to shrink to 500\(?\)

Step-by-Step Solution

Answer

The time it will take for the population of 8000 bacteria to shrink to 500 at a rate of 0.14 is approximately 27.77 hours.

Step 1: Identify the values provided

From the problem, we have that the rate of decline \(r = 0.14\), the initial bacteria population \(P = 8000\), and the reduced bacteria population \(A = 500\).

Step 2: Substitution into the formula

Substitute those values into the given formula \(H = \left(1/r\right)(\ln P-\ln A) = \left(1/0.14\right) (\ln 8000 - \ln 500)\)

Step 3: Solving the equation

First evaluate the difference in natural logarithms, then divide by the rate using a calculator. It will return the results in hours.

Key Concepts

Logarithmic Function

The logarithmic function is an essential mathematical concept, especially when dealing with exponential growth or decay in real-world situations.
It is often used to transform exponential equations into linear ones, making them easier to handle.
For example, in our exercise about bacteria population modeling, the natural logarithm (\(\ln\)) allows us to work with large numbers in a more manageable form.
By using the expression \(\ln P - \ln A\), we essentially calculate the logarithm of the ratio \(P/A\), highlighting changes in the population sizes.

• The natural logarithm function, denoted as \(\ln(x)\), is the inverse operation of exponentiation with the base \(e\), which is approximately 2.71828.
• Logarithms are particularly useful when comparing proportional differences.
• This function helps to "compress" ranges of numbers, making multiplicative processes additive.

Understanding logarithmic functions is crucial for anyone dealing with processes involving exponential changes.

Bacteria Population Modeling

Bacteria population modeling lets scientists and researchers predict how a population grows or shrinks over time based on various environmental factors.
In our exercise, we are considering how an antibiotic affects the bacterial population. When we model bacteria populations, we aim to understand both the initial population and future projections under specific conditions.
This information is invaluable for areas like medical biology and ecology.

• Exponential growth occurs when the bacteria population expands rapidly over time.
• Exponential decay, on the other hand, happens when the population decreases—such as when an antibiotic is effective.
• Bacteria population models can tell us how quickly a population will decline, helping optimize treatment strategies.

This type of modeling, combined with mathematical knowledge of logarithms, allows us to calculate specific metrics such as the time needed for a population to shrink by a particular percentage.

Rate of Decline

The rate of decline is an important characteristic in many scientific and mathematical domains, including bacteria population dynamics.
In our context, the rate of decline (\(r\)) signifies how fast the bacteria population is reducing due to antibiotic treatment.
Significantly, it translates exponential shrinking processes into manageable calculations.

• The rate of decline tells us the speed at which the population decreases.
• In mathematical terms, a higher rate means the population shrinks faster.
• By knowing this rate, one can predict the amount of time required for a population to decrease from one level to another, here using the given equation.

In the given exercise, the rate of decline, specified as 0.14, is used in combination with logarithms to calculate the number of hours needed for a population to drop from 8000 to 500.
This calculation aids in efficient decision-making in treatment and environmental management scenarios.