Chapter 4.4, Problem 99E

a.

To determine

Use the regression feature of a graphing utility to find a model of the $y=a\sin \left(bt+c\right)+d$ .

Answer to Problem 99E

  $a=13.341,b=0.501104,c=-2.30653,d=14.1216$

  $a=21.0946,b=0.461205,c=-1.46879,d=0.988152$

Explanation of Solution

Given information:

The table shows the average high temperatures (in degrees Fahrenheit) in Boston, Massachusetts (B), and Fairbanks, Alaska (F), for selected months in $2015$ .

  

Use the regression feature of a graphing utility to find a model of the form,

  $y=a\sin \left(bt+c\right)+d$

For each city. Let t represent the month, with $t=1$ corresponding to January.

Calculation:

Using regression feature of graphing utility for Boston,Massachsettes, the model $y=a\sin \left(bx+c\right)+d$ fits as follow.

  

The values of, $a=13.341,b=0.501104,c=-2.30653,d=14.1216$

Now for Fairbanks and ,Alaska, the model $y=a\sin \left(bx+c\right)+d$ fits as follow.

  

The values of , $a=21.0946,b=0.461205,c=-1.46879,d=0.988152$

Hence

  $a=13.341,b=0.501104,c=-2.30653,d=14.1216$ and $a=21.0946,b=0.461205,c=-1.46879,d=0.988152$

b.

To determine

Use the regression feature of a graphing utility to find a model of the $y=a\sin \left(bt+c\right)+d$ .

Answer to Problem 99E

  $\begin{array}{cccccccc}Month& februry& April& May& july& September& October& December\\ Boston,B& 1.251& 10.152& 16.759& 26.562& 24.881& 19.769& 6.977\\ Fairbanks,F& -9.973& 8.735& 16.657& 21.708& 10.344& 0.953& -15.847\end{array}$

Explanation of Solution

Given information:

The table shows the average high temperatures (in degrees Fahrenheit) in Boston, Massachusetts (B), and Fairbanks, Alaska (F), for selected months in $2015$ .

  

Use the models from part $\left(a\right)$ to estimate the monthly average high temperatures for the two cities in February, April, May, July, September, October, and December.

Calculation:

According to the model from part $\left(a\right)$ the temperature of both the cities in other months is as follows.

Hence, the temperature of both the cities are,

  $\begin{array}{cccccccc}Month& februry& April& May& july& September& October& December\\ Boston,B& 1.251& 10.152& 16.759& 26.562& 24.881& 19.769& 6.977\\ Fairbanks,F& -9.973& 8.735& 16.657& 21.708& 10.344& 0.953& -15.847\end{array}$

c.

To determine

Use a graphing utility to graph both models in the same viewing window. Compare the temperatures for the two cities.

Answer to Problem 99E

  

Explanation of Solution

Given information:

The table shows the average high temperatures (in degrees Fahrenheit) in Boston, Massachusetts (B), and Fairbanks, Alaska (F), for selected months in $2015$ .

  

Use a graphing utility to graph both models in the same viewing window. Compare the temperatures for the two cities.

Calculation:

Use a graphing utility to graph both models in the same viewing window is,

  

From the graph the maximum temperature of both the cities is same but minimum temperature has huge difference, the fluctuation in temperature of Boston city is less through out the year is less as conpared to Fairbanks.

Hence, in April,May,June both cities almost have same temperature.