Q. 29 - Precalculus Enhanced with Graphing Utilities

Question

Hurricanes are categorized using the Saffir- Simpson Hurricane Scale, with winds $111-130$ miles per hour (mph) corresponding to a category $3$  hurricane, winds $131-155$ mph corresponding to a category $4$  hurricane, and winds in excess of $155$ mph corresponding to a category
$5$ hurricane. The following data represent the number of major hurricanes in the Atlantic Basin (category $3,4$ or $5$) each decade from $1921 to 2010$.

(a) Draw a scatter diagram of the data for one period.

(b) Find a sinusoidal function of the form $y=A \sin (\omega x-\varphi )+B$ that models the data.

(c) Draw the sinusoidal function found in part (b) on the scatter diagram.

(d) Use a graphing utility to find the sinusoidal function of best fit.

(e) Draw the sinusoidal function of best fit on a scatter diagram of the data.

Step-by-Step Solution

Answer

(a) The scatter diagram for the data of one period is:

(b) The sinusoidal function for the given data is $H=8.5\sin (\frac{\mathrm{\pi }}{2}x-\frac{3\mathrm{\pi }}{2})+24.5$

(c) The graph of the sinusoidal function found in part (b) Is:

(d) The sinusoidal function of best fit by using the graphing utility is $y=9.46\sin (1.25x+2.91)+24.09$

(e) The sinusoidal function of best fit on the scatter diagram is:

Part (a) Step 1. Given

A table of information 

To draw a scatter diagram of the data for one period.

Part (a) Step 2. Plot the points in the graph

The decade $1$ represents $1921-1930$, similarly the next decade.

So the scatter diagram is as follows:

Part (b) Step 1. To find the amplitude A, vertical shift B of the function

Amplitude$=\frac{Largest value-Smallest value}{2}$

$$\begin{gathered} =\frac{33-16}{2} \\ =\frac{17}{2} \\ =8.5 \end{gathered}$$

Vertical shift$=\frac{Largest value+Smallest value}{2}$

$$\begin{gathered} =\frac{33+16}{2} \\ =\frac{49}{2} \\ =24.5 \end{gathered}$$

Part (b) Step 2. Determine ω

The data repeats for every $4$ decades, so,

$$\begin{gathered} T=4 \\ \frac{2\mathrm{\pi }}{\omega }=4 \\ \omega =\frac{2\mathrm{\pi }}{4} \\ =\frac{\mathrm{\pi }}{2} \end{gathered}$$

The function is $y=8.5\sin \left(\frac{\mathrm{\pi }}{2}x\right)+24.5$

Step 3. Determine horizontal shift

To determine the horizontal shift, we use the period $T=365$ and divide the interval $(0,4)$ into four sub intervals of length $4÷4=1$

$(0,1),(1,2),(2,3),(3,4)$

The sine curve is increasing on the interval $(0,1)$ and is decreasing on the interval $(1,3)$, so a local maximum occurs at $x=1$. The data indicate that a maximum occurs at $x=4$, so we must shift the graph of the function  $3$units to the right by replacing $x$ by $x-3$

Part (b) Step 3. Plot the function

So we obtain the function,

$y=8.5\sin \left(\frac{\mathrm{\pi }}{2}\right(x-3\left)\right)+24.5$

The function is:
$y=8.5\sin (\frac{\mathrm{\pi }}{2}x-\frac{3\mathrm{\pi }}{2})+24.5$

Part (c) Step 1. Draw the sinusoidal function

The graph of the sinusoidal function $y=8.5\sin (\frac{\mathrm{\pi }}{2}x-\frac{3\mathrm{\pi }}{2})+24.5$ is:

Part (d) Step 1. Finding function by using graphing utility

By entering the data in the graphing utility, we found the sinusoidal function as,

$y=9.46\sin (1.25x+2.91)+24.09$

Part (e) Step 1: Plot the function in the scatter diagram

The sinusoidal function of best fit is: $y=9.46\sin (1.25x+2.91)+24.09$

Plot the function in the graph.