Problem 51 - A Graphical Approach to College Algebra

Question

Suppose that the concentration of a bacteria sample is \(100,000\) bacteria per milliliter. If the concentration doubles every 2 hours, how long will it take for the concentration to reach \(350,000\) bacteria per milliliter?

Step-by-Step Solution

Answer

It takes approximately 3.615 hours for the concentration to reach 350,000 bacteria per milliliter.

Step 1: Understanding the problem

We know the initial concentration of bacteria is 100,000 per milliliter, and it doubles every 2 hours. We want to find the time it will take for this concentration to reach 350,000 per milliliter.

Step 2: Define the problem in exponential growth terms

Since the concentration doubles every 2 hours, this is an example of exponential growth. The formula for exponential growth is \( N(t) = N_0 \cdot 2^{t/T} \), where \( N(t) \) is the concentration at time \( t \), \( N_0 \) is the initial concentration, and \( T \) is the doubling time.

Step 3: Substitute known values into the formula

We substitute the known values into the exponential growth formula: initial concentration \( N_0 = 100,000 \), concentration after time \( t \) is \( 350,000 \), and doubling time \( T = 2 \) hours. The equation becomes \( 350,000 = 100,000 \cdot 2^{t/2} \).

Step 4: Solve for \( t \)

First, divide both sides by 100,000: \( 3.5 = 2^{t/2} \). Now, to solve for \( t \), take the logarithm of both sides: \( \log_{2}(3.5) = \frac{t}{2} \). Thus, \( t = 2 \cdot \log_{2}(3.5) \).

Step 5: Calculate using logarithms

Using the change of base formula \( \log_{2}(3.5) = \frac{\log_{10}(3.5)}{\log_{10}(2)} \) or simply using a calculator directly, compute \( \log_{2}(3.5) \). This gives approximately 1.8074. Therefore, \( t = 2 \times 1.8074 \approx 3.6148 \) hours.

Key Concepts

Doubling Time

Doubling time is a crucial concept when dealing with exponential growth. It refers to the amount of time it takes for a quantity to double in size. In our problem, the bacteria concentration doubles every 2 hours.

Understanding doubling time helps in predicting how quickly a population can grow. With a simple formula, we can determine the future quantity if we know how frequently it doubles.

For example:

• If 100,000 bacteria double every 2 hours, then after 2 hours, there will be 200,000 bacteria.
• After another 2 hours, this will increase to 400,000 bacteria.

Doubling time is commonly seen in finance, biology, and many fields involving growth.

Logarithms

Logarithms are the mathematical tool used to solve equations involving exponential growth. They help us find unknown exponents. In this problem, we use logarithms to find the time it takes for the bacteria concentration to reach a certain level.

When we have an equation like \(3.5 = 2^{t/2}\), taking the logarithm of both sides lets us isolate \(t\).

• Using base 2 logarithms: \(\log_{2}(3.5) = \frac{t}{2}\)
• This results in \(t = 2 \cdot \log_{2}(3.5)\)

Logarithms provide a way to handle exponential numbers easily. They are useful in many areas, especially when trying to "undo" exponential operations.

Bacteria Concentration

Bacteria concentration helps us understand how densely bacteria are present in a solution. This measure is important in fields like microbiology and medicine. In the problem, we start with 100,000 bacteria per milliliter, with the focus on how it grows over time.

Understanding concentration changes due to exponential growth is essential for predicting when a bacteria population might reach a dangerous level.
For instance:

• In 2 hours, concentration doubles, so it becomes 200,000 per milliliter.
• The goal is to know when it'll hit 350,000, requiring us to solve using formulas and logarithms.

By following these steps, we can accurately predict changes and make informed decisions in scientific and real-world situations.