Problem 34 - Precalculus with Limits

Question

Bacteria Growth The number of bacteria in a culture is increasing according to the law of exponential growth. The initial population is 250 bacteria, and the population after 10 hours is double the population after 1 hour. How many bacteria will there be after 6 hours?

Step-by-Step Solution

Answer

The number of bacteria after 6 hours is calculated by: 250 * e^((ln(2) / 9) * 6)

Step 1: Set up the initial equation

Represent the problem using the formula for exponential growth. From the given, we know that population after 10 hours is double the population after 1 hour. We can represent this as: 250 * e^(10r) = 2(250 * e^(1r))

Step 2: Solve for r

Next, we simplify the equation to solve for r. After cancelling equivalent terms on both sides of the equation, we have: e^(10r) = 2 * e^(r). This simplifies further to e^(9r) = 2. To solve for r, we need to take the natural log of both sides to get 9r = ln(2) and therefore, r = ln(2) / 9.

Step 3: Calculate the population after 6 hours

Now that we have the growth rate r, we can use it to calculate the population after 6 hours. Substituting P0 = 250, r = ln(2) / 9 and t = 6 into the exponential growth formula, we get: P(6) = 250 * e^((ln(2) / 9) * 6)

Step 4: Final computation

By evaluating the above expression, we get the number of bacteria after 6 hours

Key Concepts

Bacteria Growth

Bacteria growth is an exciting topic as it beautifully illustrates the concept of exponential growth. In biology, we often encounter situations where the number of bacteria in a culture increases rapidly over time. This happens because each bacterium divides to form two new bacteria, leading to a rate of growth that is proportional to the current population.

This means that the more bacteria there are, the faster they grow!

• In the given problem, the initial population of bacteria is 250.
• After 10 hours, the population doubles compared to after 1 hour.
• This type of growth is modeled using exponential functions.

Understanding this growth pattern can help scientists estimate how quickly bacteria will multiply in a given timeframe. This is valuable in fields like medicine and environmental science.

Exponential Function

The exponential function is a key mathematical tool for modeling growth processes. An exponential function is expressed in the form \( y = a \cdot e^{rt} \), where:

• \( y \) is the final quantity.
• \( a \) is the initial amount (in this case, 250 bacteria).
• \( r \) is the growth rate.
• \( t \) represents time.

In the exercise, we use this formula to describe how the bacteria population changes over time.

By substituting known values, we set up an equation to find the growth rate \( r \). The doubling of the population after 10 hours compared to 1 hour provides critical information that allows us to solve for \( r \) using the properties of exponential functions.

This understanding is crucial for predicting how populations will behave in changing environments.

Natural Logarithm

The natural logarithm, symbolized as \( \ln \), is essential for solving exponential equations. It serves as the inverse operation to the exponential function and is based on the constant \( e \), approximately 2.71828.

In the context of our exercise:

• We used \( \ln \) to isolate \( r \) in the equation \( e^{9r} = 2 \).
• This gives us \( 9r = \ln(2) \).
• Therefore, \( r = \frac{\ln(2)}{9} \).

By taking the natural log of both sides, we can effectively work backwards to find unknown variables within exponential growth problems. This step is indispensable when calculations involve the base \( e \), providing a way to solve equations that would otherwise be complex and challenging.