Biology A strain of E-coli Beu 397-recA441 is placed into a...

Biology A strain of E-coli Beu 397-recA441 is placed into a nutrient broth at $30^{\circ}$ Celsius and allowed to grow. The following data are collected. Theory states that the number of bacteria in the petri dish will initially grow according to the law of uninhibited growth. The population is measured using an optical device in which the amount of light that passes through the petri dish is measured. $$ \begin{array}{|cc|} \hline \text { Time (hours), } x & \text { Population, } \boldsymbol{y} \\ \hline 0 & 0.09 \\ 2.5 & 0.18 \\ 3.5 & 0.26 \\ 4.5 & 0.35 \\ 6 & 0.50 \\ \hline \end{array} $$ (a) Draw a scatter diagram treating time as the independent variable. (b) Using a graphing utility, build an exponential model from the data. (c) Express the function found in part (b) in the form $N(t)=N_{0} e^{k t}$ (d) Graph the exponential function found in part (b) or (c) on the scatter diagram. (e) Use the exponential function from part (b) or (c) to predict the population at $x=7$ hours. (f) Use the exponential function from part (b) or (c) to predict when the population will reach 0.75

Biology A strain of E-coli Beu 397-recA441 is placed into a nutrient broth at $30^{\circ}$ Celsius and allowed to grow. The following data are collected. Theory states that the number of bacteria in the petri dish will initially grow according to the law of uninhibited growth. The population is measured using an optical device in which the amount of light that passes through the petri dish is measured.
$$
\begin{array}{|cc|}
\hline \text { Time (hours), } x & \text { Population, } \boldsymbol{y} \\
\hline 0 & 0.09 \\
2.5 & 0.18 \\
3.5 & 0.26 \\
4.5 & 0.35 \\
6 & 0.50 \\
\hline
\end{array}
$$
(a) Draw a scatter diagram treating time as the independent variable.
(b) Using a graphing utility, build an exponential model from the data.
(c) Express the function found in part (b) in the form $N(t)=N_{0} e^{k t}$
(d) Graph the exponential function found in part (b) or (c) on the scatter diagram.
(e) Use the exponential function from part (b) or (c) to predict the population at $x=7$ hours.
(f) Use the exponential function from part (b) or (c) to predict when the population will reach 0.75

Key Concepts

Scatter Diagram

A scatter diagram is a graphical representation used to display values for two variables in a dataset. In this context, time is plotted on the horizontal axis and the population on the vertical axis. This visualization helps in observing trends, such as the exponential increase in population, and in assessing whether an exponential model is appropriate for the data.

Exponential Growth

Exponential growth is a process in which the size of a population increases at a rate proportional to its current size. This concept is fundamental in both biology and mathematics, where the population is described by models such as N(t) = N?e^(kt). Here, N? represents the initial population, and k is the constant rate of growth, meaning the growth accelerates over time if k is positive.

Exponential Model Construction

Constructing an exponential model involves fitting an exponential function, like N(t) = N?e^(kt), to the observed data. This typically involves plotting the data, using software or graphing utilities to perform regression analysis, and determining the best estimates for the parameters N? and k that describe the underlying growth phenomenon.

Parameter Estimation

Parameter estimation involves determining the values of N? and k in the exponential model that best represent the observed data. This is commonly done using curve fitting or regression techniques, which adjust the parameters to minimize the difference between the observed values and the values predicted by the model.

Predictive Modeling

Predictive modeling refers to the use of mathematical functions, such as an exponential growth model, to forecast future outcomes based on current and historical data. Once an accurate model is established, it can be used to predict future population sizes at given times or determine the time required to reach a specific population threshold.

Transcript

00:01 In this problem, we are learning how to manipulate exponential and logarithmic functions in an implied problem.

00:10 So in this case, we have some sort of population.

00:13 I think it was bacteria, which is a very straightforward and common problem for exponential learning.

00:20 So in this case, we're just going to be understanding the relationship between logarithmic and exponential models.

00:28 So for part a, we're told to make a scatter plot of the data, and that's what this would look like.

00:36 This is a scatter plot, which you can plug into a calculator or a graphem utility.

00:42 For part b, we're told, well, what's going to be the exponential equation for that scatter plot? well, when you graph it, that utility, whether it be a calculator or an online graphing utility, would give you the function.

00:57 And the function for that scatter plot is y equals 0 .093 times 1 .3384 raised to the x.

01:07 So we can definitely see that we have an exponential function because we have something raised to the x.

01:15 So for part c, we have to rearrange it to get our function into more of an explicit form.

01:23 So the new form we need is n of t equals n -n -not times e raised to the, the kt.

01:29 This is much more of a common way to write exponential functions because this gives us more information.

01:37 So we're going to let a be n -not and bx be e -raised to the k -t.

01:44 So we can see from this form that b to the x equals e -to -the -x raised to the t.

01:53 So remember, we're going to be substituting from our original equation.

01:58 So a is going to be 0903, b times e to the x is going to be 1 .3384, and we want k.

02:08 So k is going to be equivalent to the natural log of 1 .3384, which is 0 .2915.

02:17 Now this step might have been confusing.

02:19 Why am i solving for k and why do i have a natural log? k is the variable that we're missing right now in the form that we need it to be in.

02:28 And then we wanted to get rid of this e to the x.

02:32 And the way to do that is to essentially inverse it.

02:35 Use the inverse to cancel it out...

Please give Ace some feedback