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Chapter 4: Problem 41

Bacteria culture A culture of the bacterium Rhodobacter sphaeroides initially has 25 bacteria and \(t\) hours later increases at a rate of 3.4657\(e^{0.1386 t}\) bacteria per hour. Find the population size after four hours.

Short Answer

Expert verified
The population of bacteria after four hours is 44.

Step by step solution

01

Understand the Rate of Change

The rate of change of the bacteria population is given as \(3.4657e^{0.1386t}\) bacteria per hour. This is an exponential growth model, where the growth factor is related to the expression \(e^{0.1386t}\).
02

Integral Setup

To find the total population after a certain time, integrate the rate of change from the initial time 0 to the desired time, which is 4 hours. We integrate the expression as follows:\[\text{Population}(t=4) = \int_{0}^{4} 3.4657e^{0.1386t} \, dt\]
03

Perform Integration

The integral of \(3.4657e^{0.1386t}\) is calculated by applying the integration rule for exponential functions. This results in:\[\text{Population} = \left[\frac{3.4657}{0.1386}e^{0.1386t} \right]_0^4\]Calculating the constant, \( \frac{3.4657}{0.1386} \approx 25.0007\).
04

Evaluate the Definite Integral

Substitute \(t=4\) and \(t=0\) into the integrated result:\[= 25.0007 \times e^{0.1386 imes 4} - 25.0007 \times e^{0.1386 imes 0}\]\[= 25.0007 \times e^{0.5544} - 25.0007 \times e^{0}\]
05

Calculate the Exponential Terms

Calculate \(e^{0.5544} \approx 1.7408\) and since \(e^0 = 1\):\[= 25.0007 \times 1.7408 - 25.0007 \times 1\]\[= 25.0007 \times 1.7408 - 25.0007\]
06

Calculate the Population Increase

Substitute the values to find the increase in population after 4 hours:\[= 25.0007 \times (1.7408 - 1)\]\[= 25.0007 \times 0.7408 \approx 18.52\]
07

Determine Total Population

Add the increase in population to the initial population of 25 bacteria:\[25 + 18.52 = 43.52\]Since the population must be a whole number, round this to 44.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bacteria Population Modeling
In the study of population growth, particularly in microbiology, modeling the population of bacteria is crucial. When we observe bacteria like *Rhodobacter sphaeroides*, their population dynamics often follow what is called exponential growth. This is because bacteria divide and multiply at a rate proportional to their current number. Exponential growth is characterized by a growth rate that becomes proportional to the exponential functions, in this case, expressed as:
  • The initial condition - starting with 25 bacteria.
  • The rate of change - given by the formula \(3.4657e^{0.1386t}\), which means for every increase in \(t\) (time in hours), the bacteria population grows based on this exponential factor.
  • A clear timeline - here, we are interested in finding how many bacteria are present after 4 hours.
By modeling our bacteria population with these components, we gain insights into how quick changes occur and how densely a bacterial culture can grow over time.
Exponential Functions Integration
Integration is a critical process to find the total change over an interval when dealing with rates of change, such as with exponential functions for our *Rhodobacter sphaeroides*. Integrating exponential functions involves finding the antiderivative or the original function from its derivative. In our problem, the task was to integrate:
  • The function \(3.4657e^{0.1386t}\) over the interval from \(t=0\) to \(t=4\).
  • The approach uses the rule for integrating exponential functions: if you have an integrand in the form of \(ae^{kt}\), the antiderivative is \(\frac{a}{k}e^{kt}+C\), where \(C\) is the constant of integration when considering indefinite integrals.
  • Thus, for the defined interval, it becomes necessary to evaluate the resulting function at both limits.
This integration helps bridge the gap from rate of change models to the calculation of total population at a specific point in time.
Definite Integral Evaluation
To calculate the exact number of bacteria after 4 hours, we use definite integration. A definite integral gives us the exact total change over a specific interval. Here's how it works in our scenario:
  • We derived the function \(\left[\frac{3.4657}{0.1386}e^{0.1386t} \right]_0^4\) through the integration process.
  • Evaluation involves substituting the upper and lower limits into the integrated function. \(t=4\) represents the upper limit, while \(t=0\) is the starting point.
  • The calculation involves finding \(25.0007 \times e^{0.5544}\) minus \(25.0007 \times e^{0}\), which calculates the population change from the starting to the ending point in time.
Definite integral evaluation solidifies our understanding of population dynamics, providing a concrete number which, when rounded, gives us 44 bacteria after 4 hours, combining both initial and increased amounts.

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Most popular questions from this chapter

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