Problem 37 - College Algebra

Question

For the following exercises, use this scenario: A tumor is injected with 0.5 grams of Iodine-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 signifi ant 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

Approximately 332783 bacteria after 3 hours.

Step 1: Understand the Growth Model

We are dealing with exponential growth. The population of bacteria doubles every twenty minutes, which means the growth factor over each time period is 2. Since growth is continuous, we can write an equation for the population count at any time.

Step 2: Determine the Growth Rate per Minute

Since the population doubles every 20 minutes, we need to find the equivalent growth rate for one minute. The growth factor for 20 minutes is 2, so for 1 minute, it is \( \sqrt[20]{2} \). This approximates to \(1.03526\).

Step 3: Setup the Exponential Growth Equation

Using the initial population of 1350 bacteria and the growth factor per minute, the equation becomes: \( P(t) = 1350 \times 1.03526^t \), where \( t \) is the time in minutes.

Step 4: Convert 3 Hours into Minutes

Convert 3 hours into minutes, since our growth model is minute-based. 3 hours equals 180 minutes.

Step 5: Calculate Population After 3 Hours

To find the population after 3 hours, substitute \( t = 180 \) minutes into the growth equation: \( P(180) = 1350 \times 1.03526^{180} \).

Step 6: Compute the Solution

Calculate \( 1.03526^{180} \), which approximately equals \(246.5061\). Then, multiply by the initial population: \( P(180) = 1350 \times 246.5061 \approx 332783 \).

Step 7: Round to the Nearest Whole Number

Round the population size to the nearest whole number. Thus, the population size after 3 hours is approximately 332783 bacteria.

Key Concepts

Bacteria Population

When studying bacteria, understanding their population dynamics is crucial. Bacteria often reproduce at a rapid pace, allowing their population sizes to increase significantly over short periods. For example, in our scenario, the bacteria culture started with an initial population of 1350. This number serves as the baseline from which we observe changes.

• This initial count is fundamental as it sets the ground for predicting future population sizes.
• In biological studies, these counts are typically estimated using various culturing and counting techniques.

The initial population acts as the base value in exponential growth equations. It helps us model how populations change over time, especially where doubling or other factors influence growth.

Growth Rate

The growth rate is an essential variable in population studies, especially when dealing with bacteria. It allows us to understand how quickly a population is expanding.

• A growth rate expresses how many times the population increases in a specific unit of time.
• In exponential growth, the growth rate is often given as a multiplier, meaning the factor by which the population increases per unit time.

In this exercise, our bacteria culture doubles every 20 minutes. To express this as a growth rate over 1 minute, we calculate the 20th root of 2, resulting in an approximate value of 1.03526. This rate helps us build an accurate exponential model of the population growth.

Exponential Equation

Constructing an exponential equation is a method to mathematically express the growth of populations such as the one for bacteria. It provides a framework for predicting future population sizes based on current data.

• An exponential equation takes the form: \( P(t) = P_0 \times (growth\ factor)^t \), where \( P_0 \) is the initial population, and \( t \) is time.
• In our case, the equation becomes \( P(t) = 1350 \times 1.03526^t \).

This formula allows us to calculate the population at any given time \( t \). It essentially translates biological growth processes into a concise mathematical form, aiding in various predictions and simulations.

Population Doubling Time

Population doubling time refers to how long it takes for a population to double in size. This concept is integral to understanding exponential growth’s speed and intensity.

• Doubling time offers a quick insight into the pace of population explosion.
• For the bacteria culture, the doubling time is 20 minutes, which means every 20 minutes, the quantity of bacteria doubles.

This consistent doubling provides a regular pattern of exponential growth. Having a precise doubling time helps when setting up mathematical models, as it influences the population's future size and dynamics. This information is crucial in fields like epidemiology and environmental science.