Problem 409 - Precalculus

Question

For the following exercises, use this scenario: A tumor is injected with 0.5 grams of lodine- \(125,\) which has a decay rate of 1.15\(\%\) per day. A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was 1350 bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after 3 hours?

Step-by-Step Solution

Answer

The population size after 3 hours is 691200 bacteria.

Step 1: Understand the Problem

We are given that the bacteria population doubles every 20 minutes. Initially, there are 1350 bacteria. We need to find an exponential equation representing this growth and calculate the population after 3 hours.

Step 2: Set Up the Exponential Growth Equation

The exponential growth model can be represented as \( P(t) = P_0 \times a^t \), where \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population, \( a \) is the growth factor (doubling), and \( t \) is the time in the number of periods. Here, \( P_0 = 1350 \) and the population doubles every 20 minutes, so \( a = 2 \). Since the calculation is in 20-minute intervals, \( t \) in hours becomes \( 3 \times 3 = 9 \) periods.

Step 3: Calculate the Population After 3 Hours

Plug the values into the equation. \( P(3) = 1350 \times 2^9 \). Compute \( 2^9 = 512 \), thus \( P(3) = 1350 \times 512 = 691200 \).

Step 4: Check and Round the Result

Verify the calculation and ensure that the final answer is rounded to the nearest whole number if needed. Since 691200 is already a whole number, we have our solution.

Key Concepts

Bacterial Growth

Bacterial growth is a process where the number of bacteria increases in an environment. One of the most fascinating aspects of bacterial growth is its potential for rapid increase due to exponential growth. Bacteria are unicellular organisms that can reproduce quickly under optimal conditions.

Exponential growth occurs when the growth rate of the bacterial population is proportional to its current size. This means that as the population grows, the rate of growth increases as well.

• In ideal conditions, some bacteria can double their population size in just 20 minutes.
• Such rapid reproduction can lead to significant increases in population in a very short period of time.
• It is important to control and understand bacterial growth in various fields, such as medicine, agriculture, and environmental science.

Understanding the factors that influence bacterial growth, such as temperature, nutrients, and moisture, is crucial for managing it effectively.

Population Doubling Time

Population doubling time is the period it takes for a given population to double in size. This concept is crucial in understanding how quickly a population can grow under certain conditions. For bacteria, short doubling times can lead to rapid increases in numbers.

Here's how it works:

• If a bacterial population doubles every 20 minutes, it means that in 1 hour, it undergoes three doubling periods.
• This results in exponential growth, continually increasing the population.
• Understanding doubling time helps in predicting the future size of a population after a specific period.

Knowing how to calculate and interpret doubling time is essential in fields like epidemiology, ecology, and industrial microbiology to manage and utilize bacterial populations effectively.

Exponential Equation

An exponential equation is a mathematical expression that models exponential growth or decay. In the context of bacterial growth, it represents how populations increase over time based on their doubling time.

An exponential growth equation can be expressed as:

• \[ P(t) = P_0 \times a^t \]
• Where \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population, \( a \) is the growth factor, and \( t \) is the time in specific intervals.

Consider the example of a bacterial culture that doubles every 20 minutes:

• The initial population \( P_0 \) is 1350.
• The growth factor \( a \) is 2, since it doubles.
• If you want to find the population after 3 hours, recognize that 3 hours contain 9 periods of 20 minutes each.
• This gives us the equation \( P(3) = 1350 \times 2^9 \), resulting in the significant growth to 691200 bacteria.

Mastering exponential equations allows predictions of how populations evolve, essential in planning and resources management.

Decay Rate

Decay rate is the percentage by which a quantity decreases over time, often seen in processes like radioactive decay. It contrasts with growth processes, as it leads to a reduction in number or size. In the context of the initial exercise,

• Radioactive decay, such as that of Iodine-125 used in medical applications, follows an exponential decay model.
• This involves a consistent percentage reduction over successive periods.
• A 1.15% per day decay rate means that each day, the amount of substance is reduced to approximately 98.85% of the previous day's amount.

To handle decay effectively, it's necessary to comprehend exponential decay equations, which help in determining the remaining quantity of a substance after a given period. This knowledge is vital for fields requiring meticulous timing and accuracy, such as medical treatments and nuclear science.