Chapter 1.6, Problem 24E

(a)

To determine

To find: The value of $b$.

Answer to Problem 24E

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

Explanation of Solution

Given information:

The given equation is $y=a\sin \left[b\left(t-h\right)\right]+k$.

Calculation:

Consider that period is 12 months from the equation it can be seen that frequency of the cycle is $b$ . So,

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

Therefore, the value of $b$ is $\frac{\pi }{6}$.

(b)

To determine

To find: The amplitude of $a$ related to the difference $80°-30°$.

Answer to Problem 24E

The amplitude of $a$ is 25.

Explanation of Solution

Given information:

The given equation is $y=a\sin \left[b\left(t-h\right)\right]+k$.

Calculation:

The value of $a$ can be calculated as:

  $\begin{array}{c}a=\frac{80-30}{2}\\ =\frac{50}{2}\\ =25\end{array}$

Therefore, the amplitude of $a$ is 25.

(c)

To determine

To find: The value of $k$.

Answer to Problem 24E

The value of $k$ is 55.

Explanation of Solution

Given information:

The given equation is $y=a\sin \left[b\left(t-h\right)\right]+k$.

Calculation:

The average temperature can be calculated as:

  $\begin{array}{c}k=\frac{80+30}{2}\\ =\frac{110}{2}\\ =55\end{array}$

Therefore, the value of $k$ is 55.

(d)

To determine

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

Answer to Problem 24E

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

Explanation of Solution

Given information:

The given equation is $y=a\sin \left[b\left(t-h\right)\right]+k$.

Calculation:

The function should have its minimum at $t=2$ (when temperature is $30°$ F) and its maximum at $t=8$ (when temperature is $80°F$ )

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

So, the equation is:

  $y=25\sin \left[\frac{\pi }{6}\left(t-5\right)\right]+55$

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

(e)

To determine

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

Explanation of Solution

Given information:

The given table is.

Temperature data for St. Louis
Time	Temperature
1	34
2	30
3	39
4	44
5	58
6	67
7	78
8	80
9	72
10	63
11	51
12	40

Graph:

Follow the steps given below to draw the scatter plot of the given data.

Step 1: Press $\boxed{\text{STAT}}$ key and choose Edit. Now, fill the given data under L1 and L2.

Step 2: Now, set the window at $\left[-1,13\right]$ by $\left[-28,32\right]$ . Press the keystrokes $\boxed{2\text{nd}}\boxed{\text{Y}=}\boxed{1}\boxed{\text{ON}}$ , type of scatter plot. Now, press $\boxed{\text{GRAPH}}$ and $\boxed{\text{TRACE}}$ to plot the graph of given data as shown below.

  

Figure (1)

Interpretation: From the above graph of scatter plot of the stat it can be observed that it forms a smooth curve.