Chapter 4.5, Problem 92E

a.

To determine

Answer to Problem 92E

Explanation of Solution

Given information:

The percent (in decimal form) of the moon’s face illuminated on day in the year 2014, where $x=1$ represents January 1, is shown in the table.

Create a scatter plot of the data.

Calculation:

The percent (in decimal form) of the moon’s face illuminated on day in the year 2014, where $x=1$ represents January 1, is shown in the table.

  $\begin{array}{cc}x\left(\text{dayin the year}\right)& y\left(\text{illuminating percentage}\right)\\ 1& 0.0\\ 8& 0.5\\ 16& 1.0\\ 24& 0.5\\ 30& 0.0\\ 37& 0.5\end{array}$

The scatter plot of the data is as follows:

  

Hence, the result.

b.

To determine

Trigonometric model that fits the data.

Answer to Problem 92E

  $\boxed{y=0.508\sin (0.216x-1.849)+0.539}$

Explanation of Solution

Given information: The percent (in decimal form) of the moon’s face illuminated on day in the year 2014, where $x=1$ represents January 1, is shown in the table.

Find a trigonometric model that fits the data.

Calculation:

Using sine regression, the display is obtained as follows:

  

The data can be modelled as follows,

  $\begin{array}{l}y=a\sin (bx+c)+d\\ a=0.5078669623\\ b=0.2160769469\\ c=-1.84907689\\ d=0.5391240503\end{array}$

That is trigonometric model for the given data is given by,

  $y=0.508\sin (0.216x-1.849)+0.539$

Hence, $\boxed{y=0.508\sin (0.216x-1.849)+0.539}$

c.

To determine

Add the graph.

How well does the model fit the data?

Answer to Problem 92E

  

the model fits the data well.

Explanation of Solution

Given information:

The percent (in decimal form) of the moon’s face illuminated on day in the year 2014, where $x=1$ represents January 1, is shown in the table.

Add the graph of your model in part (b) to the scatter plot. How well does the model fit the data?

Calculation:

By adding the graph of above model $y=0.508\sin (0.216x-1.849)+0.539$ to the scatter plot in part a, you get,

  

Hence, from the graph above, we conclude that the model fits the data well.

d.

To determine

What is the period of the model?

Answer to Problem 92E

  $\boxed{29}$ days.

Explanation of Solution

Given information:

The percent (in decimal form) of the moon’s face illuminated on day in the year 2014, where $x=1$ represents January 1, is shown in the table.

What is the period of the model?

Calculation:

The period of the model can be determined by

  $\begin{array}{l}b=\begin{array}{c}2\pi \\ p\end{array}\\ =\begin{array}{c}2\pi \\ 0.216\end{array}\\ \approx 29\end{array}$

That is, period of the model is around $\boxed{29}$ days.

e.

To determine

Percent of the moon’s face illuminated on March 12, 2014.

Answer to Problem 92E

  $\boxed{0.657}$

Explanation of Solution

Given information:

The percent (in decimal form) of the moon’s face illuminated on day in the year 2014, where $x=1$ represents January 1, is shown in the table.

Estimate the percent of the moon’s face illuminated on March 12, 2014.

Calculation:

As $x=1$ for January 1, 2014, you get

for $x=71$ March 12, 2014

Substituting $x=71$ in $y=0.508\sin [0.216\left(x\right)-1.849]+0.539$

  $\begin{array}{l}y=0.508\sin [0.216\left(71\right)-1.849]+0.539\\ =0.508\sin [15.336-1.849]+0.539\\ =0.508\sin [13.487]+0.539\\ =0.6574781649\\ \approx 0.657\end{array}$

Hence, illumination of the moon on March 12, 2014 will be $\boxed{0.657}$ .