Chapter 4.5, Problem 87E

(a)

To determine

To find a scanner plot of data.

Answer to Problem 87E

Graph

Explanation of Solution

Given Information : $\begin{array}{cc}Day,x& Percent,y\\ 10& 0.0\\ 16& 0.5\\ 24& 1.0\\ 32& 0.5\\ 39& 0.0\\ 46& 0.5\end{array}$

Calculations:

The scatter plot & model will be:

  

The model is a good fit for the actual data.

(b)

To determine

To determine trigonometric model for the data.

Answer to Problem 87E

y = $0.5\cos \left(\frac{\pi x}{15}-5.027\right)+0.5$

Explanation of Solution

Given Information : $\begin{array}{cc}Day,x& Percent,y\\ 10& 0.0\\ 16& 0.5\\ 24& 1.0\\ 32& 0.5\\ 39& 0.0\\ 46& 0.5\end{array}$

Calculations:

Use a cosine function in the form:

y = a cos (bx - c) + d

To determine a (amplitude), find the difference of the highest & lowest value of y & divide by 2:

a = $\frac{(1-0)}{2}=0.5$

Since we are dealing with days, let period be 30 (30 days)

Since period = $\frac{2\pi }{b}$ = 30, then b = $\frac{\pi }{15}$

Consider to be the left endpoint, $\frac{c}{b}$ = x = 24, where the maximum illumination occurs. Therefore, $\frac{c}{\frac{\pi }{15}}$ = 24. c = 5.027.

To determine d, the vertical shift, find the average of the highest & lowest value of I:

d = $\frac{(1-0)}{2}=0.5$

The equation is:

y = $0.5\cos \left(\frac{\pi x}{15}-5.027\right)+0.5$

(c)

To determine

To find the graph of your model.

Answer to Problem 87E

Graph

Explanation of Solution

Given Information : $\begin{array}{cc}Day,x& Percent,y\\ 10& 0.0\\ 16& 0.5\\ 24& 1.0\\ 32& 0.5\\ 39& 0.0\\ 46& 0.5\end{array}$

Calculations:

The scatter plot & model will be:

  

The model is a good fit for the actual data.

(d)

To determine

To find period of model.

Answer to Problem 87E

period = 30 which corresponds to 30 days in a month

Explanation of Solution

Given Information : $\begin{array}{cc}Day,x& Percent,y\\ 10& 0.0\\ 16& 0.5\\ 24& 1.0\\ 32& 0.5\\ 39& 0.0\\ 46& 0.5\end{array}$

Calculations:

From (b), period = 30 which corresponds to 30 days in a month.

(e)

To determine

To find percent illuminate of the moon on June 21, 2017.

Answer to Problem 87E

y ˜ 0.835 = 83.5%

Explanation of Solution

Given Information : $\begin{array}{cc}Day,x& Percent,y\\ 10& 0.0\\ 16& 0.5\\ 24& 1.0\\ 32& 0.5\\ 39& 0.0\\ 46& 0.5\end{array}$

Calculations:

June 21, 2017 is the 172^ndday in 2017.

Therefore, x = 366 + 172 = 538

Using the model,

y = $0.5\cos \left(\frac{\pi \times (538)}{15}-5.027\right)+0.5$

y ˜ 0.835 = 83.5%