Chapter 9.6, Problem 20PPS

Solution

(a)

To estimate the number of bacteria at 11A.M, $p=300{\left(2.7\right)}^{0.02t}$ .

312 bacteria

Given information:

  $p=300{\left(2.7\right)}^{0.02t}$

t is the number of hours and t= 0 corresponds to 9:00A.M.

Concept used:

• Domain of a function $y=f\left(x\right)$ is the set of x values for which the function is defined.
• Range of a function $y=f\left(x\right)$ is the set of y -values that the function can take.
• x -intercept is the point where the graph of the function crosses/touches the x -axis.
• y -intercept is the point where the graph of the function crosses/touches the y -axis.

Calculation:

In order to find the number of bacteria at 11A.M, first calculating t by subtracting 9 from 11 because the initial time is 9:00A.M and then putting the value of t in the given function $p=300{\left(2.7\right)}^{0.02t}$ and solving it as shown below:

  $\begin{array}{l}p=300{\left(2.7\right)}^{0.02t}\\ t=11-9\\ t=2\\ p=300{\left(2.7\right)}^{0.02\times 2}\\ p=300{\left(2.7\right)}^{0.04}\\ p=312.2\\ p\approx 312\end{array}$

So, there will be about 312 bacteria at 11A.M.

(b)

To plot the graph of the function and name the p -intercept, and describe a reasonable domain and range for this situation, $p=300{\left(2.7\right)}^{0.02t}$ .

  $\begin{array}{l}p\text{-intercept }=300\\ \text{Domain}=\left(0,\infty \right)\\ \text{Range}=\left(300,\infty \right)\end{array}$

Given information:

  $p=300{\left(2.7\right)}^{0.02t}$

t is the number of hours and t= 0 corresponds to 9:00A.M.

Concept used:

• Domain of a function $y=f\left(x\right)$ is the set of x values for which the function is defined.
• Range of a function $y=f\left(x\right)$ is the set of y -values that the function can take.
• x -intercept is the point where the graph of the function crosses/touches the x -axis.
• y -intercept is the point where the graph of the function crosses/touches the y -axis.

Calculation:

In order to find the p -intercept putting the value of t in the function, that is t = 0 at 9:00A.M as shown below:

  $\begin{array}{l}p=300{\left(2.7\right)}^{0.02t}\\ t=0\\ p=300{\left(2.7\right)}^{0.02\times 0}\\ p=300\end{array}$

The p -intercept is 300. This represents 300 bacteria in the culture at 9:00A.M.

In order to get the graph of $p=300{\left(2.7\right)}^{0.02t}$ , first make a table by taking some random values of x and find the corresponding values of y . And then plot the ordered pairs and connect the points with a smooth curve, so the table is:

X	$p=300{\left(2.7\right)}^{0.02t}$	Y
- 2	$p=300{\left(2.7\right)}^{0.02\times -2}$	288
- 1	$p=300{\left(2.7\right)}^{0.02\times -1}$	294
0	$p=300{\left(2.7\right)}^{0.02\times 0}$	300
1	$p=300{\left(2.7\right)}^{0.02\times 1}$	306
2	$p=300{\left(2.7\right)}^{0.02\times 2}$	312

Graph of the function is shown below:

  

From the above graph it is clear that the curve intersecting the y -axis at the point (0, 300). So, the y -intercept of the function is at $y=300$ .

Since the t represents the time in hours, so it can’t be negative. Thus, the domain of the function is all real numbers greater than 0, that is, $\text{Domain}=\left(0,\infty \right)$ .

The range of the function will be all real number greater than 300, that is, $\text{Range}=\left(300,\infty \right)$ .