Calculating Bacterial Growth Time With Exponential Equation N(h) = 100e^(0.25h)

by Admin 80 views

Introduction

This article delves into the mathematical modeling of bacterial growth using an exponential function. Understanding bacterial growth is crucial in various fields, including microbiology, medicine, and environmental science. The provided equation, $N(h) = 100 e^{0.25 h}$, serves as a mathematical representation of how a bacterial population increases over time. In this model, N(h) represents the number of bacteria present after h hours, the initial population is 100, and the exponential term e raised to the power of 0.25h dictates the growth rate. The base of the natural logarithm, e, is approximately 2.71828, and it's fundamental in modeling continuous growth phenomena. The coefficient 0.25 in the exponent signifies the growth rate constant, influencing how rapidly the bacterial population expands. This exploration aims to not only understand the mechanics of the equation but also to apply it in predicting the time it takes for the bacterial population to reach a specific threshold, such as 450 bacteria.

The exponential growth model is a powerful tool, but it's essential to recognize its limitations. In real-world scenarios, bacterial growth is often constrained by factors such as nutrient availability, space limitations, and the accumulation of waste products. As a result, the exponential growth phase is typically followed by a stationary phase where growth slows down and eventually plateaus. Despite these limitations, the exponential model provides a valuable approximation of bacterial growth during the initial stages, allowing scientists and researchers to make informed predictions and decisions. Further, comprehending the dynamics of bacterial growth is not just an academic exercise; it has practical implications in controlling bacterial infections, optimizing industrial fermentation processes, and managing environmental concerns related to bacterial contamination. This article will guide you through the process of solving for the time it takes for the bacterial population to reach 450, offering insights into the practical applications of exponential growth modeling.

Problem Statement

Our task is to determine the time it takes for the bacterial population to reach 450. We are given the equation $N(h) = 100 e^{0.25 h}$, where N(h) is the number of bacteria after h hours. To solve this, we need to find the value of h when N(h) equals 450. This involves setting up the equation and using logarithmic properties to isolate h. This mathematical process not only provides a specific answer but also enhances our understanding of how exponential functions are used to model real-world phenomena. The steps we take to solve this problem highlight the relationship between exponential growth and logarithmic scales, a fundamental concept in various scientific disciplines. Moreover, the practical application of this mathematical exercise underscores the importance of quantitative analysis in biological studies. By understanding the underlying mathematical principles, we can apply similar approaches to model other growth phenomena, such as population dynamics in ecology or compound interest in finance.

Solving for Time

To find the time h when the number of bacteria N(h) reaches 450, we set up the equation:

450=100e0.25h450 = 100 e^{0.25 h}

First, divide both sides by 100:

4.5=e0.25h4.5 = e^{0.25 h}

Next, take the natural logarithm (ln) of both sides to eliminate the exponential term:

ln(4.5)=ln(e0.25h)ln(4.5) = ln(e^{0.25 h})

Using the property of logarithms, $ln(e^x) = x$, we get:

ln(4.5)=0.25hln(4.5) = 0.25 h

Now, divide both sides by 0.25 to solve for h:

h=ln(4.5)0.25h = \frac{ln(4.5)}{0.25}

Using a calculator, we find that $ln(4.5) ≈ 1.5041$, so:

h≈1.50410.25h ≈ \frac{1.5041}{0.25}

h≈6.0164h ≈ 6.0164

Therefore, it will take approximately 6.02 hours for the bacterial population to reach 450.

This step-by-step solution demonstrates the power of mathematical tools in predicting real-world outcomes. The use of logarithms to solve exponential equations is a fundamental technique in many scientific and engineering fields. The precision in this calculation is important, especially in applications where timing is critical, such as in medical treatments or industrial processes. Furthermore, this process highlights the importance of understanding the properties of logarithms and exponential functions, which are essential for modeling and analyzing various natural phenomena.

Conclusion

In conclusion, using the exponential growth model $N(h) = 100 e^{0.25 h}$, we determined that it takes approximately 6.02 hours for the bacterial population to reach 450. This calculation underscores the utility of mathematical models in predicting biological phenomena. The exponential growth model, while a simplification of real-world bacterial growth, provides a valuable tool for understanding and estimating population dynamics during the initial phases of growth. The ability to manipulate and solve such equations is a crucial skill in many scientific disciplines.

Moreover, the process of solving for the time it takes for the bacterial population to reach a specific number illustrates the importance of understanding the properties of exponential functions and logarithms. These mathematical concepts are fundamental not only in biology but also in various other fields, including physics, chemistry, finance, and engineering. By mastering these concepts, scientists and researchers can develop a deeper understanding of the world around them and make more informed decisions. The example provided in this article serves as a practical illustration of how mathematical models can be applied to solve real-world problems, highlighting the interconnectedness of mathematics and science. As we continue to explore and understand complex systems, the ability to model and predict outcomes using mathematical tools will remain essential.