Chapter 0.6, Problem 44E

To determine

To calculate: The sinusoid equation $y=A\cos \left(Bx\right)+C$ that models the temperature cycle in Steamboat Springs.

Answer to Problem 44E

The obtained function is $y=-26.85\cos \left(\frac{\pi }{6}x\right)+\text{55}\text{.75}$ .

Explanation of Solution

Given Information:

The table is defined as,

JAN	FEB	MAR	APR	MAY	JUN	JULY	AUG	SEP	OCT	NOV	DEC
$28.9$	$33.8$	$42.0$	$53.6$	$65.4$	$75.5$	$82.6$	$80.3$	$72.5$	$60.4$	$43.3$	$30.7$

Formula Used:

The period is defined as,

  $P=\left|\frac{2\pi }{B}\right|$

The general function is defined as,

  $y=A\cos \left(Bx\right)+C$

Where, $A$ is amplitude, $C$ is vertical shift.

Calculation:

Consider the given table,

Rewrite the table and take Jan as 0 and Dec as 11.

0	1	2	3	4	5	6	7	8	9	10	11
$28.9$	$33.8$	$42.0$	$53.6$	$65.4$	$75.5$	$82.6$	$80.3$	$72.5$	$60.4$	$43.3$	$30.7$

Draw all the points on the plane.

  

It can be observed form the graph that, it is trigonometric graph.

It is given that the temperature range from a $28.9$ in January to a high of $82.6$ . So find the amplitude.

  $\begin{array}{c}A=\left|\frac{82.6-28.9}{2}\right|\\ =\left|\frac{53.7}{2}\right|\\ =26.85\end{array}$

As the rotation of the graph of the function opposite side . So the amplitude will not be negative.

  $A=-26.85$

Now, find the value of $B$ .

As the total period covered by the graph is 12 points. So $P=12$ .

  $\begin{array}{c}12=\frac{2\pi }{B}\\ B=\frac{2\pi }{12}\\ =\frac{\pi }{6}\end{array}$

Now, find the value the vertical displacement $C$ .

As the initial point of the displacement is $19.75$ . Now, the maximum point on the given graph is 80.

  $\begin{array}{c}C=82.6-26.85\\ =\text{55}\text{.75}\end{array}$

Substitute all the values in the formula.

  $y=-26.85\cos \left(\frac{\pi }{6}x\right)+\text{55}\text{.75}$

Therefore, the obtained function is $y=-26.85\cos \left(\frac{\pi }{6}x\right)+\text{55}\text{.75}$ .