Chapter 4.6, Problem 83E

a.

To determine

The period of each function.

Answer to Problem 83E

  $\boxed{12}$ months.

Explanation of Solution

Given information:

The normal monthly high temperatures $H$ (in degrees Fahrenheit) in Erie, Pennsylvania, are approximated by

  $H(t)=56.94-20.86\cos (\begin{array}{c}\pi t\\ 6\end{array})-11.58\sin (\begin{array}{c}\pi t\\ 6\end{array})$

and the normal monthly low temperatures $L$ are approximated by

  $L(t)=41.80-17.13\cos (\begin{array}{c}\pi t\\ 6\end{array})-13.39\sin (\begin{array}{c}\pi t\\ 6\end{array})$

where is $t$ the time (in months), with $t=1$ corresponding to January (see figure).

(a) What is the period of each function?

Calculation: The normal monthly high temperatures $H$ (in degrees Fahrenheit) and low temperatures $L$ in Erie, Pennsylvania, are approximated by

  $H(t)=56.94-20.86\cos (\begin{array}{c}\pi t\\ 6\end{array})-11.58\sin (\begin{array}{c}\pi t\\ 6\end{array})$

  $L(t)=41.80-17.13\cos (\begin{array}{c}\pi t\\ 6\end{array})-13.39\sin (\begin{array}{c}\pi t\\ 6\end{array})$

where is $t$ the time.

Let us consider the following figure,

  

From the above figure, we can see that the curve for normal monthly high temperatures $H$ (in degrees Fahrenheit) and low temperatures $L$ starts from $0$ till $12$ months.

Hence, the period of $H(t)$ and $L(t)$ will be $\boxed{12}$ months.

b.

To determine

Greatest and smallest difference between the normal high and normal low temperatures.

Answer to Problem 83E

Greatest in summer months and smallest in winter months.

Explanation of Solution

Given information:

The normal monthly high temperatures $H$ (in degrees Fahrenheit) in Erie, Pennsylvania, are approximated by

  $H(t)=56.94-20.86\cos (\begin{array}{c}\pi t\\ 6\end{array})-11.58\sin (\begin{array}{c}\pi t\\ 6\end{array})$

and the normal monthly low temperatures $L$ are approximated by

  $L(t)=41.80-17.13\cos (\begin{array}{c}\pi t\\ 6\end{array})-13.39\sin (\begin{array}{c}\pi t\\ 6\end{array})$

Where is $t$ the time (in months), with $t=1$ corresponding to January (see figure).

During what part of the year is the difference between the normal high and normal low temperatures greatest? When is it smallest?

Calculation:

The normal monthly high temperatures $H$ (in degrees Fahrenheit) and low temperatures $L$ in Erie, Pennsylvania, are approximated by

  $H(t)=56.94-20.86\cos (\begin{array}{c}\pi t\\ 6\end{array})-11.58\sin (\begin{array}{c}\pi t\\ 6\end{array})$

  $L(t)=41.80-17.13\cos (\begin{array}{c}\pi t\\ 6\end{array})-13.39\sin (\begin{array}{c}\pi t\\ 6\end{array})$

where is $t$ the time.

Let us consider the following figure,

  

From the above figure, we can see that, the difference between the normal high $H$ and normal low temperatures $L$ is greatest in summer months.

Also, the difference between the normal high $H$ and normal low temperatures $L$ is smallest in winter months.

Hence, the difference between the temperatures is greatest in summer months and smallest in winter months.

c.

To determine

Approximate the lag time.

Answer to Problem 83E

  $\boxed{0.5}$ months

Explanation of Solution

Given information:

The normal monthly high temperatures $H$ (in degrees Fahrenheit) in Erie, Pennsylvania, are approximated by

  $H(t)=56.94-20.86\cos (\begin{array}{c}\pi t\\ 6\end{array})-11.58\sin (\begin{array}{c}\pi t\\ 6\end{array})$

and the normal monthly low temperatures $L$ are approximated by

  $L(t)=41.80-17.13\cos (\begin{array}{c}\pi t\\ 6\end{array})-13.39\sin (\begin{array}{c}\pi t\\ 6\end{array})$

where is $t$ the time (in months), with $t=1$ corresponding to January (see figure).

The sun is northernmost in the sky around June 21, but the graph shows the warmest temperatures at a later date. Approximate the lag time of the temperatures relative to the position of the sun.

Calculation: The normal monthly high temperatures $H$ (in degrees Fahrenheit) and low temperatures $L$ in Erie, Pennsylvania, are approximated by

  $H(t)=56.94-20.86\cos (\begin{array}{c}\pi t\\ 6\end{array})-11.58\sin (\begin{array}{c}\pi t\\ 6\end{array})$

  $L(t)=41.80-17.13\cos (\begin{array}{c}\pi t\\ 6\end{array})-13.39\sin (\begin{array}{c}\pi t\\ 6\end{array})$

where is $t$ the time.

Let us consider the following figure,

  

From the above graph, it can be seen thst the curves attains its peak in the $7^{th}$ month and that is in th month of july.

So, the warmest of the temperature will be in the month of July, instead of being on 21 of June., which means approximately $0.5$ months later.

Hence, the lag time of the temperatures relative to the position of sun will be $\boxed{0.5}$ months.