Chapter 4.6, Problem 83E

a.

To determine

What is the period of each function?

Answer to Problem 83E

  $\boxed{12months}$

Explanation of Solution

Given information:

The normal monthly high temperatures H (in degrees Fahrenheit) in Erie, Pennsylvania, are approximated by $\text{H(t) = 57}\text{.54 - 18}\text{.53 cos}\frac{\pi t}{6}-14.03\sin \frac{\pi t}{6}$ and the normal monthly low temperatures $L$ are approximated by $L\left(t\right)=42.03-15.99\cos \frac{\pi t}{6}-14.32\sin \frac{\pi t}{6}$ where t is the time (in months), with $t=1$ corresponding to January (see figure). (Source: NOAA)

  

What is the period of each function?

Calculation:

Period $H\left(f\right)$ = $12months$

Period $L\left(f\right)$ = $12months$

Hence, the Period of each function= $12months$

b.

To determine

What is the maximum and least temperature?

Answer to Problem 83E

In July that means in summer difference is max. Is least in fall season.

Explanation of Solution

Given information:

The normal monthly high temperatures H (in degrees Fahrenheit) in Erie, Pennsylvania, are approximated by $\text{H(t) = 57}\text{.54 - 18}\text{.53 cos}\frac{\pi t}{6}-14.03\sin \frac{\pi t}{6}$ and the normal monthly low temperatures $L$ are approximated by $L\left(t\right)=42.03-15.99\cos \frac{\pi t}{6}-14.32\sin \frac{\pi t}{6}$ where t is the time (in months), with $t=1$ corresponding to January (see figure). (Source: NOAA)

  

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

Calculation:

In July that means in summer difference is max. Is least in fall season.

Hence the difference is max. In July and least in fall season.

c.

To determine

Approximate the lag time of the temperatures relative to the position of the sun.

Answer to Problem 83E

Graph shows warmest temperature in July starting lag time of $\frac{1}{2}month$ .

Explanation of Solution

Given information:

The normal monthly high temperatures H (in degrees Fahrenheit) in Erie, Pennsylvania, are approximated by $\text{H(t) = 57}\text{.54 - 18}\text{.53 cos}\frac{\pi t}{6}-14.03\sin \frac{\pi t}{6}$ and the normal monthly low temperatures $L$ are approximated by $L\left(t\right)=42.03-15.99\cos \frac{\pi t}{6}-14.32\sin \frac{\pi t}{6}$ where t is the time (in months), with $t=1$ corresponding to January (see figure). (Source: NOAA)

  

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:

Graph shows warmest temperature in July starting lag time of $\frac{1}{2}month$ .

Hence , the warmest temperature in july starting lag time of $\frac{1}{2}month$ .