Chapter 10, Problem 3TP

Solution

(a)

To find: Make a table of the usage of gas foe every month.

The table is given.

Given information:

The given data is below:

  

Given that temperature for January is taken as $t=0$ .

The consumption of gas is denoted by $y_2$ and average temperature is denoted by $y_1$.

The table is given below:

Temperature(y1)	0	1	2	3	4	5	6	7	8	9	10	11
Gas consumption(y2)	20,000	27000	23000	22000	21000	14000	8000	9000	13000	15000	19000	23000

(b)

To find: Use the values of in part $(a)$ draw the graph and find the trigonometric model for the given values.

The model of the trigonometric function is $\begin{array}{l}{\text{y}}_{\text{2}}=\text{7500sin}\left(\frac{\pi }{\text{5}}{\text{y}}_{\text{1}}\text{-1}\right)+\text{17500}\\ \end{array}$ .

Given information:

The given data is below:

Temperature(y1)	0	1	2	3	4	5	6	7	8	9	10	11
Gas consumption(y2)	20,000	27000	23000	22000	21000	14000	8000	9000	13000	15000	19000	23000

The graph using the above data is below:

  

The model of sinusoidal equation is:

  $\begin{array}{l}\text{y}=\text{asin(bx}+\text{c)}+\text{d}\\ \end{array}$

  $a$ is amplitude. It is calculated with the formula $\left|a\right|=\frac{M-m}{2}$ .

Here , $M$ is the maximum value and $m$ is minimum value .

From the graph it is clear that $M=27000,m=8000$

  $\begin{array}{l}\left|a\right|=\frac{27000-8000}{2}\\ \left|a\right|=\frac{15000}{2}\\ \left|a\right|=7500\end{array}$

Period of the graph is $P=10$ .

The vertical shift is given by the formula $k=\frac{M+m}{2}$ .

  $\begin{array}{l}d=\frac{M+m}{2}\\ d=\frac{27000+8000}{2}\\ d=\frac{35000}{2}\\ d=17500\end{array}$

Apply the formula $P=\frac{2\pi }{b}$ .

Substitute the value of $P$ gives:

  $\begin{array}{l}10=\frac{2\pi }{b}\\ b=\frac{2\pi }{10}\\ b=\frac{\pi }{5}\end{array}$

Horizontal shift is given as $1$ as graph shifted from $\left(0,0\right)$ to $\left(1,0\right)$ .

The model of the function is given by substituting the above found value in $\begin{array}{l}\text{y}=\text{asin(bx}+\text{c)}+\text{d}\\ \end{array}$ .

  $\begin{array}{l}{\text{y}}_{\text{2}}=\text{7500sin}\left(\frac{\pi }{\text{5}}{\text{y}}_{\text{1}}\text{-1}\right)+\text{17500}\\ \end{array}$

This is the required function.

(b)

To find: Graph the two regression equation in the same plane.

The graph is given.

Given information:

The given data is below:

Temperature(y1)	0	1	2	3	4	5	6	7	8	9	10	11
Gas consumption(y2)	20,000	27000	23000	22000	21000	14000	8000	9000	13000	15000	19000	23000

The given data is fit into the sinusoidal equation then regression equation becomes:

  $y_2=7904.568\sin (0.6283y_1+0.421)+16992.9356$

The graph of the equation $y_2=7904.568\sin (0.6283y_1+0.421)+16992.9356$ and $\begin{array}{l}{\text{y}}_{\text{2}}=\text{7500sin}\left(\frac{\pi }{\text{5}}{\text{y}}_{\text{1}}\text{-1}\right)+\text{17500}\\ \end{array}$ is given below:

  

It is observed that the graph $\begin{array}{l}{\text{y}}_{\text{2}}=\text{7500sin}\left(\frac{\pi }{\text{5}}{\text{y}}_{\text{1}}\text{-1}\right)+\text{17500}\\ \end{array}$ is shifted to the right side of the $y$ axis .