Problem 13 - Algebra and Trigonometry

Question

An infectious strain of bacteria increases in number at a relative growth rate of 200% per hour. When a certain critical number of bacteria are present in the bloodstream, a person becomes ill. If a single bacterium infects a person, the critical level is reached in 24 hours. How long will it take for the critical level to be reached if the same person is infected with 10 bacteria?

Step-by-Step Solution

Answer

It takes approximately 22.85 hours to reach the critical level with 10 bacteria.

Step 1: Understand the Growth Model

The growth of bacteria in this problem can be described using an exponential model: \[ N(t) = N_0 e^{rt} \]where \( N(t) \) is the number of bacteria at time \( t \), \( N_0 \) is the initial number of bacteria, \( r \) is the growth rate, and \( t \) is time. Given that the growth rate is 200%, \( r = 2 \).

Step 2: Model the Critical Condition for Single Bacterium

If a single bacterium (\( N_0 = 1 \)) leads to the critical level in 24 hours, we first calculate the critical number of bacteria \( N_c \):\[ N_c = 1 \, e^{2 \times 24} \].

Step 3: Calculate the Critical Number of Bacteria

Solve for \( N_c \) when \( t = 24 \) hours:\[ N_c = e^{48} \]This represents the number of bacteria required to reach the critical level.

Step 4: Model the New Initial Infection

When a person is infected with 10 bacteria, the initial number of bacteria \( N_0 = 10 \). We need to find the time \( t' \) at which the bacteria count reaches \( N_c \):\[ e^{48} = 10 \, e^{2t'} \].

Step 5: Solve for New Time t'

To find \( t' \), take the natural logarithm of both sides:\[ \ln(e^{48}) = \ln(10) + 2t' \]This simplifies to:\[ 48 = \ln(10) + 2t' \].

Step 6: Calculate t'

Rearrange the equation and solve for \( t' \):\[ 2t' = 48 - \ln(10) \]\[ t' = \frac{48 - \ln(10)}{2} \]Use a calculator to find \( t' \).

Step 7: Final Calculation

Calculate \( t' \) using \( \ln(10) \approx 2.302 \):\[ t' = \frac{48 - 2.302}{2} \approx \frac{45.698}{2} \approx 22.849 \text{ hours} \].

Key Concepts

Relative Growth Rate

The concept of relative growth rate is crucial when studying processes like bacterial infections, where populations grow rapidly. In mathematics, the growth rate of a population is the rate at which it increases over time. For this exercise, the bacteria have a relative growth rate of 200% per hour.

• This means the number of bacteria doubles every hour.
• A 200% growth rate can be represented by a growth constant of 2 in the exponential growth formula.

These high growth rates are typical of bacterial populations under optimal conditions but understanding this rate helps us model how a population will expand over time using exponential functions.

Initial Number

The initial number of bacteria, denoted as \( N_0 \), is the quantity of bacteria present at the start of our observation. It is an important factor in predicting how quickly a bacterial infection can become problematic. In this example:

• The initial number \( N_0 \) for a single bacterium scenario is 1.
• For a scenario with 10 bacteria, \( N_0 \) is 10.

Starting with a higher number of bacteria results in the critical threshold being reached faster, due to the rapid nature of exponential growth. Understanding the starting point of growth provides insight into how quickly an infection can escalate and allows effective interventions.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is an important concept when dealing with exponential equations, especially in growth models. It helps in solving for time \( t \) when the exponential function's output is known.In mathematical terms, the natural logarithm is the power to which \( e \) (approximately 2.718) must be raised to yield a given number. In our bacterial growth context:

• We use \( \ln \) to isolate time \( t' \) when solving \( N(t) = e^{48} \).
• The equation \( \ln(e^{48}) = \ln(10) + 2t' \) relies on the property that \( \ln(e^x) = x \).

Utilizing \( \ln \) allows us to linearize the exponential equation and easily solve for the desired value.

Bacterial Infection

Bacterial infections can spread rapidly based on the exponential growth characteristic of bacteria. This concept is reflected in scenarios where a small initial infection can lead to a large population of bacteria over a seemingly short period. Understanding bacterial growth allows us to model how an infection might progress:

• As bacteria multiply, they reach a threshold (or critical level) that can cause illness.
• Different initial conditions, such as starting with 1 or 10 bacteria, impact how quickly this threshold is reached.

By understanding how bacterial infections proliferate, we can predict outcomes and plan medical treatments and interventions more effectively.

Critical Level

A critical level in the context of infections, refers to the specific number of bacteria that lead to visible symptoms or illness in a host. Once this level is reached, medical symptoms begin to present themselves. In this exercise:

• The critical level is reached when the count \( N(t) = e^{48} \).
• This mathematical representation helps establish when a person becomes ill.

Critical levels are essential in medical monitoring and planning, allowing healthcare professionals to prepare for symptomatic outbreaks and apply timely interventions.