Problem 47 - Precalculus Mathematics for Calculus

Question

Bacteria Growth A bacteria culture contains 1500 bacteria initially and doubles every hour. (a) Find a function that models the number of bacteria after \(t\) hours. (b) Find the number of bacteria after 24 hours.

Step-by-Step Solution

Answer

The function is \( N(t) = 1500 \cdot 2^t \). After 24 hours, there are 25,165,824,000 bacteria.

Step 1: Understanding the Growth Model

The bacteria population doubles every hour, which is an exponential growth situation. An exponential growth model generally has the form \( N(t) = N_0 \cdot 2^{(t/T)} \), where \(N_0\) is the initial amount of bacteria, \(t\) is the time in hours, and \(T\) is the doubling period.

Step 2: Setting Up the Function

For this problem, the initial population \(N_0\) is 1500 and the doubling period \(T\) is 1 hour. Thus, the function modeling the bacteria population is \( N(t) = 1500 \cdot 2^t \).

Step 3: Calculating the Population After 24 Hours

To find the number of bacteria after 24 hours, substitute \(t = 24\) into the model: \( N(24) = 1500 \cdot 2^{24} \).

Step 4: Performing the Calculation

Calculate \( 2^{24} \), which is 16,777,216. Then multiply this by the initial amount, 1500: \( N(24) = 1500 \times 16,777,216 = 25,165,824,000 \).

Key Concepts

Doubling Time

Doubling time is a fundamental concept in understanding exponential growth. It refers to the period it takes for a quantity to double in size or amount. In the context of bacteria population modeling, doubling time gives us insight into how quickly the bacterial count increases. For bacteria doubling every hour, the doubling time is precisely one hour. This predictable growth pattern is not only fascinating but also crucial for planning and managing biological experiments.
If you know the initial population, you can calculate the population at any given time using the doubling time. For example, if you start with 1500 bacteria and the doubling time is 1 hour, the population will be:

• After 1 hour: 1500 x 2 = 3000
• After 2 hours: 3000 x 2 = 6000
• After 3 hours: 6000 x 2 = 12000

The concept of doubling time is very useful when considering things like bacteria cultures in labs or understanding how quickly an infection can spread. It helps us predict future growth and make informed decisions.

Exponential Functions

Exponential functions are mathematical models that describe a constant percentage growth or decay over equal time intervals. These functions are generally represented in the form of \( N(t) = N_0 \times a^t \), where:

• \(N(t)\) is the quantity at time \(t\).
• \(N_0\) is the initial quantity.
• \(a\) is the base of the exponential, representing the growth factor.
• \(t\) is the time changes the population experiences.

In the context of our exercise, the base \(a\) is 2, indicating that the quantity doubles with each unit of time. Exponential functions are key to understanding phenomena such as compound interest in finance, radioactive decay in physics, and changes in populations in biology.
The important factor in exponential functions is their rapid growth rate. This makes them appropriate models for phenomena that grow quickly over time. In our bacterial example, every hour the population was effectively multiplied by 2. Such a model allows us to quickly compute populations after many doubling periods, leading to vast numbers.

Bacteria Population Modeling

Bacteria population modeling is a practical application of exponential growth. It allows scientists to predict how a bacterial culture will grow over time given an initial amount and a known doubling rate. Through modeling, important strategies can be developed for health, industrial, and environmental applications.
The formula \( N(t) = N_0 \times 2^t \) captures this exponential growth perfectly. In the case of our bacteria exercise, 1500 bacteria grow into a multitude over a short period, specifically doubling every hour. This exponential model becomes vital when considering scenarios such as:

• Testing the efficiency of antibiotics.
• Maintaining controlled bacterial cultures for biotechnology industries.
• Understanding the spread of bacterial infections in public health.

Being able to predict these growth patterns helps in planning interventions and understanding the nature of bacterial colonization. Exponential models make calculations manageable, even as the numbers grow to immense scales, like calculating how 1500 bacteria can expand to over 25 billion in just 24 hours.