Chapter 5.9, Problem 1AYU

(a)

To determine

To find: The scatter diagram treating time as a independent variable.

Answer to Problem 1AYU

The scatter diagram treating time as independent variable.

Explanation of Solution

Given:

The given table is shown in table 1

Table 1

Time hours x	Population y
0	0.09
2.5	0.18
3.5	0.26
4.5	0.35
6	0.50

Calculation:

Consider the time as independent variable.

The scatter plot for the functions in table 1 is shown in Figure 1

  

Figure 1

(b)

To determine

To find: The exponential model from the given data.

Answer to Problem 1AYU

The exponential function is $y=0.09e^{0.277x}$ .

Explanation of Solution

From the graph shown in Figure 1, the exponential function with the help of the graphing utility is,

  $y=0.09e^{0.277x}$ .

(c)

To determine

To find: The function from the exponential model in the form of $N\left(t\right)=N_0{e}^{kt}$ .

Answer to Problem 1AYU

The required model is $N\left(t\right)=0.09e^{0.277t}$ .

Explanation of Solution

Given:

The exponential function is $y=0.09e^{0.277x}$ .

Calculation:

Consider the given model is,

  $N\left(t\right)=N_0{e}^{kt}$

Compare the given function with the given model as,

  $N\left(t\right)=0.09e^{0.277t}$

(d)

To determine

To find: The graph of the exponential function from part (b) or (c) on the scatter diagram.

Answer to Problem 1AYU

The required plot is shown in Figure 2

Explanation of Solution

Given:

The exponential function is $y=0.09e^{0.277x}$ and $N\left(t\right)=0.09e^{0.277t}$ .

Calculation:

The required plot is shown in Figure 2

  

Figure 2

(e)

To determine

To find: The graph of the exponential function from part (b) or (c) to predict the population at $x=7\text{hours}$ .

Answer to Problem 1AYU

The population is 0.63.

Explanation of Solution

Given:

The exponential function is $N\left(t\right)=0.09e^{0.277t}$ .

Calculation:

Consider the given function is,

  $N\left(t\right)=0.09e^{0.277t}$

Then,

  $\begin{array}{c}N\left(7\right)=0.09e^{0.277\left(7\right)}\\ =0.63\end{array}$

(f)

To determine

To find: The graph of the exponential function from part (b) or (c) to time in which the population will reach 0.75.

Answer to Problem 1AYU

The desired population is reached at $7.65\text{hour}$ .

Explanation of Solution

Given:

The exponential function is $N\left(t\right)=0.09e^{0.277t}$ .

Calculation:

Consider the given function is,

  $0.75=0.09e^{0.277t}$

Then,

  $\begin{array}{l}0.75=0.09e^{0.277\left(t\right)}\\ t=7.65\text{hour}\end{array}$