Chapter 1.6, Problem 83E

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

$H\left(t\right)=57.54-18.53\text{cos}\frac{\pi t}{6}-14.03\text{sin}\frac{\pi t}{6}$

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

$L\left(t\right)=42.03-15.99\text{cos}\frac{\pi t}{6}-14.32\text{sin}\frac{\pi t}{6}$

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

(a) What is the period of each function $?$

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

(c) 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.