Problem 35 - College Algebra with Modeling and Visualization

Question

Find \(C\) and a so that \(f(x)=C a^{x}\) models the situation described. State what the variable \(x\) represents in your formula. (Answers may vary.) There are initially 5000 bacteria, and this sample doubles in size every hour.

Step-by-Step Solution

Answer

The function is \(f(x) = 5000 \cdot 2^x\), where \(x\) is time in hours.

Step 1: Define Variable x

In this situation, the variable \(x\) represents time in hours. This is because the rate of doubling mentioned in the problem occurs every hour.

Step 2: Determine the Initial Quantity, C

The initial number of bacteria is given as 5000. In exponential functions of the form \(f(x) = C a^x\), \(C\) represents the initial quantity. Therefore, \(C = 5000\).

Step 3: Establish the Base, a

The bacteria double every hour, meaning after 1 hour the population is twice as large. This can be expressed as \(a = 2\), given that for each increment of \(x\) by 1, the population is multiplied by 2.

Step 4: Write the Exponential Function

Using \(C = 5000\) and \(a = 2\), the function becomes \(f(x) = 5000 \cdot 2^x\). This formula models the situation where the bacteria population starts at 5000 and doubles every hour.

Key Concepts

Initial Quantity

In exploring the growth model of our bacterial population, understanding the concept of **initial quantity** is essential. The initial quantity in an exponential growth function is the starting value before any growth has occurred. Think of it as the population before time starts ticking.
For our exercise, the initial quantity is the number of bacteria present at the very beginning, which amounts to 5000. This quantity is not just a random number—it's your starting point in the exponential growth equation. The exponential function typically takes the form \( f(x) = C \cdot a^x \), where \( C \) denotes this initial quantity.
Recognizing the initial quantity is crucial because it anchors the entire equation, providing the baseline from which all growth is measured. So, in the equation \( f(x) = 5000 \cdot a^x \), the 5000 represents the initial bacteria count.

Exponential Function

An **exponential function** is a special type of mathematical function used to model situations where quantities grow rapidly over time. This isn't just any growth; it's multiplicative, meaning that the quantity increases by a constant factor in each time period.
For our scenario, the exponential function is defined as \( f(x) = 5000 \cdot 2^x \). Let's break it down:

• \( f(x) \) represents the number of bacteria after \( x \) hours.
• \( 5000 \) is our initial quantity (from the initial session).
• \( 2 \) is the base of this exponential function, indicating the doubling factor or growth rate.

The function is powerful because it allows us to see how the bacteria count balloons over time. For every hour that passes, \( x \) increases by 1, making the population double due to the base \( a = 2 \). In real-world terms, using this function helps predict future amounts of a growing population or quantity.

Population Doubling

One of the most fascinating aspects of exponential growth is **population doubling**. This refers to the time it takes for a given population to double in size, thanks to consistent and predictable growth rates.
In our bacteria example, the population doubles every hour. This means that every hour, the bacteria count multiplies by 2, which is captured in the equation \( a = 2 \). Here’s how this works:

• After 1 hour, the initial 5000 bacteria become 10,000.
• After 2 hours, those 10,000 bacteria double again, reaching 20,000.
• This doubling keeps repeating for each hour, demonstrating rapid growth.

Understanding this concept is useful not just in biology but across various fields like finance, physics, or any area involving rapid growth. It highlights that in exponential growth, even small time periods can result in significant increases, showcasing the power of compounding in nature and mathematics.