CPM Homework Help : PC Problem 7-84

The average high temperature for Glen Rock, PA can be modeled sinusoidal with time. The highest average temperature of $85^\circ\text{F}$ occurs on July 17 (Day $197$ ). The lowest average temperature of $37^\circ\text{F}$ occurs on January 18 (Day $17$ ). Find an equation that will model the temperature in Glen Rock, PA. Use the equation to estimate the high temperature on Valentines Day (Day $44$ ). Assume $365$ days in a year.

Step 1:

Plot the given points for the highest temperature and the lowest temperature.

Step 2:

Half of the vertical distance between the low point and the high point is the amplitude.

Step 3:

Identify the location of the midline. This is the vertical shift.

Step 4:

Double the $x$ -distance between the low point and the high point for the period. Then calculate $b=\frac{2\pi}{p}$ .

Step 5:

Using a cosine function, the horizontal shift is the horizontal distance from the $y$ -axis to the first high point.

Step 6:

$f(x) = a\cos\left(b\left(x − h\right)\right) + k$