Chapter 0.6, Problem 24E

(a)

To determine

To calculate : The value of $b$ , assuming that the period is $12$ months.

Answer to Problem 24E

The value of $b$ is $\frac{\pi }{6}$ .

Explanation of Solution

Given information:

The table given below gives the average monthly temperatures in St. Louis for a $12$ -month period starting with January. Model the monthly temperature with an equation of the form: $y=a\sin (b(t-h))+k$ ,with $y$ in degrees Fahrenheit and $t$ in months.

Time (Months)	Temperature $\left(°F\right)$
$\begin{array}{l}1\\ 2\\ 3\\ 4\\ 5\\ 6\end{array}$	$\begin{array}{l}34\\ 30\\ 39\\ 44\\ 58\\ 67\end{array}$
$\begin{array}{l}7\\ 8\\ 9\\ 10\\ 11\\ 12\end{array}$	$\begin{array}{l}78\\ 80\\ 72\\ 63\\ 51\\ 40\end{array}$

Calculation:

It is given that period is $12$ months and from the equation it can be seen that frequency of the cycle is $b$ .

Hence,

  $\begin{array}{l}b=\frac{2\pi }{12}\\ =\frac{\pi }{6}\end{array}$

(b)

To determine

To calculate : Relation between the amplitude $a$ to the difference $80°-30°$ .

Answer to Problem 24E

Relation between the amplitude $a$ to the difference $80°-30°$ is $25$ .

Explanation of Solution

Given information:

The table given below gives the average monthly temperatures in St. Louis for a $12$ -month period starting with January. Model the monthly temperature with an equation of the form: $y=a\sin (b(t-h))+k$ ,with $y$ in degrees Fahrenheit and $t$ in months.

Time (Months)	Temperature $\left(°F\right)$
$\begin{array}{l}1\\ 2\\ 3\\ 4\\ 5\\ 6\end{array}$	$\begin{array}{l}34\\ 30\\ 39\\ 44\\ 58\\ 67\end{array}$
$\begin{array}{l}7\\ 8\\ 9\\ 10\\ 11\\ 12\end{array}$	$\begin{array}{l}78\\ 80\\ 72\\ 63\\ 51\\ 40\end{array}$

Calculation:

It is half of the difference,

So $a=\frac{80-30}{2}=25$

(c)

To determine

To calculate : The value of $k$ using the value of b found in part (a).

Answer to Problem 24E

The value of $k$ is $55$ .

Explanation of Solution

Given information:

The table given below gives the average monthly temperatures in St. Louis for a $12$ -month period starting with January. Model the monthly temperature with an equation of the form: $y=a\sin (b(t-h))+k$ ,with $y$ in degrees Fahrenheit and $t$ in months.

Time (Months)	Temperature $\left(°F\right)$
$\begin{array}{l}1\\ 2\\ 3\\ 4\\ 5\\ 6\end{array}$	$\begin{array}{l}34\\ 30\\ 39\\ 44\\ 58\\ 67\end{array}$
$\begin{array}{l}7\\ 8\\ 9\\ 10\\ 11\\ 12\end{array}$	$\begin{array}{l}78\\ 80\\ 72\\ 63\\ 51\\ 40\end{array}$

Calculation:

It is the average of the minimum and maximum temperatures.

So,

  $\begin{array}{l}k=\frac{80+30}{2}\\ =55\end{array}$

(d)

To determine

To calculate : The value of $h$ and write an equation for $y$ .

Answer to Problem 24E

The value of $h$ is $5$ and the equation for $y$ is $25\sin \left[\frac{\pi }{6}\left(t-5\right)\right]+55$ .

Explanation of Solution

Given information:

The table given below gives the average monthly temperatures in St. Louis for a $12$ -month period starting with January. Model the monthly temperature with an equation of the form: $y=a\sin (b(t-h))+k$ ,with $y$ in degrees Fahrenheit and $t$ in months.

Time (Months)	Temperature $\left(°F\right)$
$\begin{array}{l}1\\ 2\\ 3\\ 4\\ 5\\ 6\end{array}$	$\begin{array}{l}34\\ 30\\ 39\\ 44\\ 58\\ 67\end{array}$
$\begin{array}{l}7\\ 8\\ 9\\ 10\\ 11\\ 12\end{array}$	$\begin{array}{l}78\\ 80\\ 72\\ 63\\ 51\\ 40\end{array}$

Calculation:

The function should have its minimum at $t=2$ (when the temperature is maximum) at $t=8$ (when the temperature is ${80}^{0}F$ )

Hence the value

  $\begin{array}{l}h=\frac{2+8}{2}\\ =5\end{array}$

Hence the equation can be written as $y=25\sin \left[\frac{\pi }{6}\left(t-5\right)\right]+55$ .

(e)

To determine

To graph : The equation for $y$ on a scatter plot of the data.

Explanation of Solution

Given information:

The table given below gives the average monthly temperatures in St. Louis for a $12$ -month period starting with January. Model the monthly temperature with an equation of the form: $y=a\sin (b(t-h))+k$ ,with $y$ in degrees Fahrenheit and $t$ in months.

Time (Months)	Temperature $\left(°F\right)$
$\begin{array}{l}1\\ 2\\ 3\\ 4\\ 5\\ 6\end{array}$	$\begin{array}{l}34\\ 30\\ 39\\ 44\\ 58\\ 67\end{array}$
$\begin{array}{l}7\\ 8\\ 9\\ 10\\ 11\\ 12\end{array}$	$\begin{array}{l}78\\ 80\\ 72\\ 63\\ 51\\ 40\end{array}$

Graph:

The graph of the equation is:

  

Interpretation:

The domain is $\left[-1,13\right]$ and the range is $\left[-10,100\right]$ .