Bacteria Growth: Time To Reach 2300 Population?

by Alex Johnson 48 views

Have you ever wondered how quickly a population of bacteria can grow? It's a fascinating topic, especially when you can use mathematical models to predict their growth. In this article, we'll dive into a specific scenario where the bacteria population, denoted as P(h), increases exponentially over time h (measured in hours). We'll use the function P(h) = 1800e^(0.18h) to model this growth, and our main goal is to figure out how long it will take for the population to reach 2300. So, let's put on our math hats and explore this bacterial growth problem together!

Understanding Exponential Growth

Before we jump into the calculations, let's take a moment to understand what exponential growth really means. Exponential growth is a type of growth where the rate of increase becomes more rapid in proportion to the growing total number or size. In simpler terms, the bigger the population, the faster it grows. This is often seen in biological populations like bacteria, where each bacterium can divide and multiply, leading to a rapid increase in numbers.

In our case, the function P(h) = 1800e^(0.18h) describes this exponential growth. Let's break down the components:

  • P(h): This represents the population of bacteria at a given time h (in hours). It's what we're trying to find in this problem – specifically, the time h when P(h) reaches 2300.
  • 1800: This is the initial population of bacteria at time h = 0. It's the starting point of our growth curve.
  • e: This is the base of the natural logarithm, an important mathematical constant approximately equal to 2.71828. It's the heart of exponential functions.
  • 0.18: This is the growth rate constant. It determines how quickly the population increases over time. A higher growth rate constant means faster growth.
  • h: This is the time in hours, our independent variable. We'll be solving for h in this problem.

So, the function tells us that the population starts at 1800 and grows exponentially with a rate constant of 0.18. Now that we understand the function, let's get to the problem at hand: finding the time it takes for the population to reach 2300.

Solving for Time: How Long to Reach 2300?

Our mission is to find the time h when the bacteria population P(h) reaches 2300. To do this, we'll set P(h) equal to 2300 in our equation and solve for h. Here's the equation:

2300 = 1800e^(0.18h)

This looks a bit intimidating, but don't worry, we'll break it down step by step. Our goal is to isolate h on one side of the equation. To do this, we'll use some algebraic techniques, including logarithms.

  1. Divide both sides by 1800: This will get the exponential term by itself.

    2300 / 1800 = e^(0.18h)

    This simplifies to:

    1. 2778 = e^(0.18h)

  2. Take the natural logarithm of both sides: The natural logarithm (ln) is the inverse function of the exponential function with base e. This means that ln(e^x) = x. Taking the natural logarithm of both sides will help us get rid of the exponential term.

    ln(1.2778) = ln(e^(0.18h))

    Using the property of logarithms, this becomes:

    ln(1. 2778) = 0.18h

  3. Solve for h: Now we have a simple equation to solve for h. Just divide both sides by 0.18.

    h = ln(1.2778) / 0.18

Now, we can use a calculator to find the value of ln(1.2778) and then divide by 0.18. Make sure your calculator is in radian mode when calculating natural logarithms.

ln(1. 2778) ≈ 0.2453

h ≈ 0.2453 / 0.18

h ≈ 1.363 hours

So, it will take approximately 1.363 hours for the bacteria population to reach 2300. But wait, the question asks us to round our answer to the nearest hour. So, let's do that.

Rounding the Answer

We found that h is approximately 1.363 hours. To round this to the nearest hour, we look at the decimal part. Since 0.363 is less than 0.5, we round down to 1 hour.

Therefore, it will take approximately 1 hour for the bacteria population to reach 2300.

Conclusion

We've successfully calculated the time it takes for a bacteria population to reach a certain size, given its exponential growth function. We learned how to use the natural logarithm to solve for time in an exponential equation. The key steps involved understanding the exponential growth function, setting up the equation, using logarithms to isolate the variable, and finally, solving for the time. Remember, exponential growth can be very rapid, as we saw in this example where a population grows significantly in just a little over an hour!

This exercise demonstrates the power of mathematical models in understanding and predicting real-world phenomena, such as population growth. By understanding these concepts, we can gain insights into various biological, financial, and other systems that exhibit exponential behavior.

For further exploration of exponential growth and related concepts, you can visit websites like Khan Academy's Exponential Growth and Decay section. This will provide you with more examples, exercises, and a deeper understanding of this fascinating mathematical topic. Now you have a solid understanding of how to calculate the time it takes for bacterial populations to grow, and you're equipped to tackle similar problems in the future!