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Chapter 4: Problem 22

Bacterial Growth. Suppose that a biologist performs an experiment in which he or she measures the rate at which a specific type of bacterium reproduces asexually in different culture media. The experiment shows that in Medium A the bacteria reproduce once every 60 minutes, and in Medium B the bacteria reproduce once every 90 minutes. Assume that a single bacterium is placed on each culture medium at the beginning of the experiment. Write a program that calculates and plots the number of bacteria present in cach culture at intervals of 3 hours from the beginning of the experiment antil 24 hours have elapsed. Make two plots, one a linear xy plot and the other a linear-log (semi logy) plot. How do the numbers of bacteria compare on the two media after 24 hours?

Short Answer

Expert verified
In this experiment, the bacteria growth in Medium A and Medium B can be calculated using the formula N(t) = N0 * a^n. For Medium A, with a reproduction time of 60 minutes, there are 3 intervals in 3 hours (180 minutes). For Medium B, with a reproduction time of 90 minutes, there are 2 intervals in 3 hours. By calculating the number of bacteria at each 3-hour interval, we can create linear xy and linear-log (semi logy) plots to compare the numbers of bacteria in each medium after 24 hours. Note that you will need to use a software or programming language capable of mathematical computations and data visualization, such as Python, MATLAB, or Excel, to perform the calculations and create the plots.

Step by step solution

01

Determine bacterial growth formula

The formula we will use to calculate bacterial growth is N(t) = N0 * a^n, where N(t) represents the number of bacteria after a certain time (t), N0 is the initial number of bacteria, a is the growth factor, and n is the number of intervals. For Medium A, the growth factor will be 2 (since the bacteria reproduce once every 60 minutes, doubling their number), and for Medium B, the growth factor will be 2 as well (as they reproduce once every 90 minutes). Since we have to calculate the number of bacteria at intervals of 3 hours, we need to calculate the number of intervals that match this criteria. We can do this by dividing the total time (3 hours or 180 minutes) by the time required for 1 reproduction.
02

Calculate the number of intervals

For Medium A, we have a reproduction time of 60 minutes. To calculate the 3-hour intervals, we can divide 180 minutes by 60 minutes. Number of intervals in Medium A = 180 / 60 = 3 For Medium B, we have a reproduction time of 90 minutes. To calculate the 3-hour intervals, we can divide 180 minutes by 90 minutes. Number of intervals in Medium B = 180 / 90 = 2
03

Calculate the number of bacteria at each time interval

Now, we can calculate the number of bacteria at each time interval for both Medium A and Medium B using the bacterial growth formula. For example, after the first 3 hours, we can calculate the number of bacteria as follows: Medium A: N(t) = 1 * 2^3 = 8 Medium B: N(t) = 1 * 2^2 = 4 Similarly, we can calculate the number of bacteria at each time interval (every 3 hours) until 24 hours have passed.
04

Plot the results

We can now create the plots using the calculated bacteria count at each 3-hour interval. For the linear xy plot, the x-axis will represent the time (in hours), and the y-axis will represent the number of bacteria. For the linear-log (semi logy) plot, the x-axis will represent the time, and the y-axis will represent the logarithm of the number of bacteria.
05

Compare the numbers of bacteria after 24 hours

By analyzing the linear xy plot and the linear-log (semi logy) plot after 24 hours, we can compare the number of bacteria in Medium A and Medium B and deduce how they differ. The comparison can be drawn by looking at the number of bacteria in each medium after 24 hours. NOTE: While we have provided the necessary steps to solve and plot the bacterial growth for both Medium A and Medium B, you will need to do the calculations and create the plot using a software or programming language capable of handling mathematical computations and data visualization, such as Python, MATLAB, or Excel.

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Most popular questions from this chapter

Program 1sgfit from Example \(4.7\) required the user to specify the number of input data points before entering the values. Modify the program so that it reads an arbitrary number of data values using a while loop and stops reading input values when the user presses the Enter key without typing any values. Test your program using the same two data sets that were used in Example 4,7. (Hint: The input function returns an empty array (1) if a user presses Enter without supplying any data. You can use function isempty to test for an empty array and stop reading data when one is detected.)

Harmonic Mean. The harmonic mean is yet another way of calculating a mean for a set of numbers. The harmonic mean of a set of numbers is given by the equation $$ \text { harmonic mean }=\frac{N}{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\ldots+\frac{1}{x_{N}}} $$ Write a MATLAB program that will read in an arbitrary number of positive input values and calculate the harmonic mean of the numbers. Use any method that you desire to read in the input values. Test your program by calculating the harmonic mean of the four numbers \(10,5,2\), and \(5 .\)

Program doy in Example \(4.3\) calculates the day of year associated with any given month, day, and year. As written, this program docs not check to sce if the data entered by the user is valid. It will accept nonsense values for months and days and do calculations with them to produce meaningless results. Modify the program so that it checks the input values for validity before using them. If the inputs are invalid, the program should tell the user what is wrong and quit. The year should be a number greater than zero, the month should be a number between 1 and 12 , and the day should be a number between 1 and a maximum that depends on the month. Use a switch construct to implement the bounds checking performed on the day.

Fibonacei Numbers. The nth Fibonacci number is defined by the following recursive equations: $$ \begin{aligned} &f(1)=1 \\ &f(2)=2 \\ &f(\mathrm{n})=f(\mathrm{n}-1)+f(\mathrm{n}-2) \end{aligned} $$ Therefore, \(f(3)=f(2)+f(1)=2+1=3\), and so forth for higher numbers, Write an M-file to calculate and write out the nth Fibonacci number for \(n>2\), where \(n\) is input by the user. Use a while loop to perform the calculation.

Geometric Mean. The geometric mean of a set of numbers \(x_{1}\) through \(x_{n}\) is defined as the \(n\)th root of the product of the numbers $$ \text { geometric mean }=\sqrt[4]{x_{1} x_{2} x_{3} \ldots x_{n}} $$ Write a MATLAB program that will accept an arbitrary number of positive input values and calculate both the arithmetic mean (i.e., the average) and the geometric mean of the numbers. Use a while loop to get the input values and terminate the inputs when a user enters a negative number. Test your program by calculating the average and geometric mean of the four numbers \(10,5,2\), and 5 .
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