Chapter 4.5, Problem 99E

Solution

(a)

To Graph: To graph the scatter plot of the given data.

Given information: Given moon’s face illuminated in percentage with different days.

Graph:

  

Interpretation:

Scatter plot of the data can be clearly seen through graph.

(b)

To Find: To find a trigonometric model for the given data.

Given information: Given moon’s face illuminated in percentage with different days.

Calculation:

We use a cosine function in the form of $y=p\cos (qx-r)+s$

Now, we have to find the value of $p,q,r\text{and}s$ which is calculated below:

  $\begin{array}{l}p=\frac{\text{Differnce}\text{of}\text{the}\text{highest}\text{and}\text{lowest}\text{value}\text{of}y}{2}\\ p=\frac{1-0}{2}=\frac{1}{2}=0.5\\ \text{Since,}x\text{represent}\text{the}\text{number}\text{of}\text{days}\\ \text{Therefore,}\text{we}\text{let}\text{period}\text{for}\text{30}\text{days}\\ \therefore qx=2\pi \\ \text{put}x=30\\ 30q=2\pi \\ q=\frac{2\pi }{30}=\frac{\pi }{15}\end{array}$

Also, the maximum illumination occurs first is in the 10^th day

  $\begin{array}{l}\text{Therefore,}\\ qx-r=0\\ \text{put}x=10andq=\frac{\pi }{15}\\ 10\times \frac{\pi }{15}-r=0\\ r=\frac{2\pi }{3}\end{array}$

And

  $\begin{array}{l}s=\text{average}\text{of}\text{highest}\text{and}\text{lowest}\text{value}\\ s=\frac{1+0}{2}=\frac{1}{2}=0.5\end{array}$

Thus, the required equation is $0.5\cos \left(\frac{\pi x}{15}-\frac{2\pi }{3}\right)+0.5$

(c)

To Graph: To graph the above model in the scatter plot.

Given information: Given model $0.5\cos \left(\frac{\pi x}{15}-\frac{2\pi }{3}\right)+0.5$

Graph:

  

Explanation:

In the graph it can be seen that the model is a good fit for the actual data.

(d)

To Find: To find the period of the given model.

Given information: Given moon’s face illuminated in percentage with different days with scatter and model graph.

Explanation:

Since xrepresents the number of days and also in the calculation of model above the period is determined is 30 which represent the number of days in a month.

Therefore, the period is 30.

(e)

To Find: To find the estimated illumination of the moon on June 28, 2021.

Given information: Given model $0.5\cos \left(\frac{\pi x}{15}-\frac{2\pi }{3}\right)+0.5$

Explanation:

Since June 28, 2021 is the 179^th day in 2021.

Therefore,

  $\begin{array}{l}x=366+179\\ x=545\end{array}$

Now, using the equation we have

  $\begin{array}{l}0.5\cos \left(\frac{\pi x}{15}-\frac{2\pi }{3}\right)+0.5\\ \text{put}x=545\text{then}\\ \text{0}\text{.5cos}\left(\frac{545\pi }{15}-\frac{2\pi }{3}\right)+0.5\\ 0.5\cos \left(\frac{109\pi }{3}-\frac{2\pi }{3}\right)+0.5\\ 0.5\cos \left(\frac{107\pi }{3}\right)+0.5\\ 0.5\times 0.5385+0.5\\ 0.2693+0.5\\ 0.77\text{approx}\end{array}$

Thus, the estimated illumination of the moon is 77%.