Question

The growth of Mycobacterium tuberculosis bacteria can be modeled by the function \(N(t)=a e^{0.166 t}\), where \(N\) is the number of cells after \(t\) hours and \(a\) is the number of cells when \(t=0\). a. At 1:00 P.M., there are \(30 \mathrm{M}\). tuberculosis bacteria in a sample. Write a function that gives the number of bacteria after 1:00 P.M. b. Use a graphing calculator to graph the function in part (a). c. Describe how to find the number of cells in the sample at 3:45 P.M.

Short answer

a) The function that gives the number of bacteria after 1:00 P.M. is \(N(t) = 30 e^{0.166 t}\). b) To graph this function, use a graphing calculator or tool and set \(y = N(t)\) and \(x = t\). c) The number of cells at 3:45 P.M. is found by substituting \(t = 2.75\) into the function and solving: \(N(2.75) = 30 e^{0.166 \times 2.75}\)

Step-by-step solution

Form the function

The function of bacteria growth of Mycobacterium tuberculosis is given by \(N(t)=a e^{0.166 t}\). It is stated that at 1:00 P.M., there are 30 bacteria in a sample. Because \(a\) represents the initial number of bacteria when \(t=0\), and because \(t=0\) is our starting point of 1:00 P.M., it can be concluded \(a=30\). Therefore, the function is \(N(t) = 30 e^{0.166 t}\).

Graphing the function

To graph this exponential growth function, use a graphing calculator or graphing tool. Due to the nature of this exercise, it's impossible to graph the function here, but you can do it yourself by setting \(y = N(t)\) and \(x = t\). This will give you the cell growth over time.

Find the number of cells at 3:45 P.M.

To determine the number of cells at 3:45 P.M., the value \(t\) should be taken as the number of hours passed since 1:00 P.M. (2.75 hours). Simply substitute \(t = 2.75\) into the function \(N(2.75) = 30 e^{0.166 \times 2.75}\) and solve to get the number of cells at 3:45 P.M.

Key concepts

Exponential Functions

Exponential functions are a type of mathematical expression where a constant base is raised to a variable exponent. This means that the rate of growth or decay changes exponentially, rather than linearly. In the context of the Mycobacterium tuberculosis growth, the function is given by \(N(t) = a e^{0.166 t}\), where \(a\) represents the initial quantity, and \(e\) is the base of the natural logarithm (approximately equal to 2.718). This type of function is often used to describe scenarios where growth accelerates over time, such as population growth, radioactive decay, and, as in this exercise, bacterial growth.

When graphed, the function produces a curve that rises steeply, showing how rapidly the number increases as time progresses. A key feature of exponential functions is that they have a constant percentage rate of change. That means no matter the starting amount, the growth pattern remains consistent.
Make sure to remember the characteristic 'J-shaped' curve in these graphs, which signifies ongoing growth or decay without reaching a plateau.

Graphing Calculators

Graphing calculators are powerful tools that help visualize complex functions like the exponential growth in this problem. They allow you to input equations and see their graphical representations, providing immediate insight into how the function behaves. For the function \(N(t) = 30 e^{0.166 t}\), you would enter it into the calculator as, for example, \(y = 30 e^{0.166 x}\), where \(x\) represents time in hours.

Using a graphing calculator, you can plot the function to observe the steep growth curve. Pay attention to the settings on your calculator. Ensure the x-axis is set to display the time intervals you're interested in, and the y-axis shows the range of bacteria population you wish to observe. This visual representation can help you understand how quickly the bacteria population increases over time, showing how the graph's steepness signifies faster growth.

Function Notation

Function notation is a systematic way to express mathematical functions, which helps you keep track of the input and output relationships. In a function like \(N(t) = a e^{0.166 t}\), \(N\) is the name of the function, \(t\) is the variable (input), and the resulting value from the expression is the output or the value of the function. The function notation clearly defines \(N(t)\) as a way to denote the number of bacteria at any given time \(t\).

Understanding this notation is crucial for substituting values and solving problems accurately. For instance, when asked to find the bacteria count at 3:45 P.M., you substitute \(t = 2.75\), calculating \(N(2.75)\) to get your result. This systematic substitution and calculation offer a structured approach to problem-solving, allowing for consistent results.

Mathematical Modeling

Mathematical modeling involves creating a mathematical representation of a real-world process. In this exercise, the growth of Mycobacterium tuberculosis is modeled using an exponential function. This model helps predict future outcomes based on current data, like finding the number of bacteria after a certain number of hours.

To create a model, you need a thorough understanding of the factors influencing the process. Here, you know the rate of growth is 0.166 per hour and the initial quantity is 30. With these parameters, you build the model \(N(t) = 30 e^{0.166 t}\), which serves as a predictive tool to estimate the bacteria count over time.

• This approach is not only used in biology but in numerous other fields like economics (for financial projections), physics (for population dynamics), and environmental studies (for resource depletion).
• Mathematical modeling helps simplify the complexities of nature into an understandable and manageable form.

These models can be adjusted as more data becomes available, ensuring they remain accurate and useful.