Chapter 6, Problem 54RE

(a)

To determine

To Draw: The scatter plot of the data for one period.

Explanation of Solution

Given: The following data represent the average monthly temperatures for Phoenix, Arizona.

  $\begin{array}{ccccccccccccc}\text{Month,}m& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12\\ \text{AMT, }T& 51& 55& 63& 67& 77& 86& 90& 90& 84& 71& 59& 52\end{array}$

Using graphing utility to plot the data point and show scatter plot.

  

(b)

To determine

To find: The sinusoidal function of the form $y=A\sin \left(\omega x-\varphi \right)+B$ that fits the data.

Answer to Problem 54RE

  $y=19.6331\sin \left(0.5145x-2.0537\right)+71.2176$

Explanation of Solution

Given: The following data represent the average monthly temperatures for Phoenix, Arizona.

  $\begin{array}{ccccccccccccc}\text{Month,}m& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12\\ \text{AMT, }T& 51& 55& 63& 67& 77& 86& 90& 90& 84& 71& 59& 52\end{array}$

Using four points from data table to find 4 equation.

For $\left(1,51\right),51=A\sin \left(\omega -\varphi \right)+B\mathrm{...}\left(i\right)$

For $\left(3,63\right),63=A\sin \left(3\omega -\varphi \right)+B\mathrm{...}\left(ii\right)$

For $\left(7,90\right),90=A\sin \left(7\omega -\varphi \right)+B\mathrm{...}\left(iii\right)$

For $\left(11,59\right),59=A\sin \left(11\omega -\varphi \right)+B\mathrm{...}\left(iv\right)$

Using above four equation to find $A,B,\omega \text{ and }\varphi$ .

So, $A=\text{19}\text{.6331,}B=\text{71}\text{.2176,}\omega =0.5145,\varphi =2.0537$

The sinusoidal function, $y=19.6331\sin \left(0.5145x-2.0537\right)+71.2176$

(c)

To determine

To draw: The sinusoidal function on scatter plot.

Explanation of Solution

Given: The following data represent the average monthly temperatures for Phoenix, Arizona.

  $\begin{array}{ccccccccccccc}\text{Month,}m& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12\\ \text{AMT, }T& 51& 55& 63& 67& 77& 86& 90& 90& 84& 71& 59& 52\end{array}$

The sinusoidal function, $y=19.6331\sin \left(0.5145x-2.0537\right)+71.2176$

  

(d)

To determine

To find: The sinusoidal function of the form $y=A\sin \left(\omega x-\varphi \right)+B$ that fits the data.

Answer to Problem 54RE

  $y=19.7448\sin \left(0.5236x-2.1543\right)+70.4167$

Explanation of Solution

Given: The following data represent the average monthly temperatures for Phoenix, Arizona.

  $\begin{array}{ccccccccccccc}\text{Month,}m& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12\\ \text{AMT, }T& 51& 55& 63& 67& 77& 86& 90& 90& 84& 71& 59& 52\end{array}$

Using graphing utility to find the best sinusoidal function of the given data.

The function is $y=19.7448\sin \left(0.5236x-2.1543\right)+70.4167$

(e)

To determine

To draw: The sinusoidal function on scatter plot of best fit.

Explanation of Solution

Given: The following data represent the average monthly temperatures for Phoenix, Arizona.

  $\begin{array}{ccccccccccccc}\text{Month,}m& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12\\ \text{AMT, }T& 51& 55& 63& 67& 77& 86& 90& 90& 84& 71& 59& 52\end{array}$

The sinusoidal best fit function, $y=19.7448\sin \left(0.5236x-2.1543\right)+70.4167$