Problem 39 - Intermediate Algebra

Question

For Problems \(39-44\), express your answers to the nearest whole number. Suppose that in a certain culture, the equation \(Q(t)=\) \(1000 e^{0.4 t}\) expresses the number of bacteria present as a function of the time \(t\), where \(t\) is expressed in hours. How many bacteria are present at the end of 2 hours? 3 hours? 5 hours?

Step-by-Step Solution

Answer

2226 at 2 hours, 3320 at 3 hours, 7389 at 5 hours.

Step 1: Understanding the function

The function given is \( Q(t) = 1000 e^{0.4t} \). Here, \( Q(t) \) represents the number of bacteria at time \( t \) in hours. \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

Step 2: Calculate bacteria count at t=2

Substitute \( t = 2 \) into the equation: \( Q(2) = 1000 e^{0.4 \times 2} = 1000 e^{0.8} \). Calculate \( e^{0.8} \), which approximately equals 2.22554. Therefore, \( Q(2) \approx 1000 \times 2.22554 \approx 2226 \).

Step 3: Calculate bacteria count at t=3

Substitute \( t = 3 \) into the equation: \( Q(3) = 1000 e^{0.4 \times 3} = 1000 e^{1.2} \). Calculate \( e^{1.2} \), which approximately equals 3.32012. Therefore, \( Q(3) \approx 1000 \times 3.32012 \approx 3320 \).

Step 4: Calculate bacteria count at t=5

Substitute \( t = 5 \) into the equation: \( Q(5) = 1000 e^{0.4 \times 5} = 1000 e^{2} \). Calculate \( e^{2} \), which approximately equals 7.38906. Therefore, \( Q(5) \approx 1000 \times 7.38906 \approx 7389 \).

Key Concepts

Bacteria Population

In the fascinating world of microbiology, understanding how bacteria grow is crucial for many applications. Bacteria populations often grow rapidly under the right conditions. This growth can be understood through mathematical expressions such as the one used in our problem. A bacteria population typically follows a pattern of exponential growth, where the number of bacteria increases at a rate proportional to the current population. This trait means that as time goes on, the population doesn't just add, but multiplies. In this exercise, the bacteria population starts with 1000 bacteria. This number serves as the initial condition or starting point of the bacterial culture. As time passes, the population grows exponentially, leading to much larger numbers over short periods. By correctly substituting time values into the growth equation, we can predict the size of the bacteria population at future time points.

Natural Exponential Function

The underlying principle of exponential growth is captured by the natural exponential function. This function involves the constant \( e \), which is approximately 2.71828. The natural exponential function is commonly used to model exponential growth situations, such as that of the bacteria.The function for our problem is \( Q(t) = 1000 e^{0.4t} \). Here, \( e \) serves as the base for this exponential function, allowing us to predict how rapidly the bacteria are growing over time. The expression \( e^{0.4t} \) indicates that as the time \( t \) increases, the exponent grows linearly, and due to the nature of \( e \), the overall function inflates exponentially.This exponential function allows precise calculation of bacteria counts as they increase over time, making it a powerful tool in mathematical modeling of real-world phenomena.

Mathematical Modeling

Mathematical modeling plays a crucial role in understanding and predicting real-world processes, such as bacterial growth. By using a mathematical model like \( Q(t) = 1000 e^{0.4t} \), we establish a relationship between time and bacterial population size.Mathematical modeling translates these natural phenomena into a calculable and predictive framework. It enables scientists and researchers to draw accurate predictions and perform what-if analyses by changing various parameters within the model. Through this model, we calculate precise outcomes like bacteria counts after certain hours, assisting with detailed insights and planning. By substituting different time values into our model, we can simulate population growth and adapt strategies accordingly, whether for laboratory experiments or managing bacteria in industrial applications.