Chapter 12.2, Problem 55E

To determine

To calculate:

Sinusoidal general function that models the temperatures, using $t=1$ to represent winter.

Answer to Problem 55E

Sinusoidal general function that models the temperatures, using $t=1$ to represent winter is $y=25\sin \left(\frac{\pi }{2}t+C\right)+61$ .

Explanation of Solution

Given information:

Winter	Spring	Summer	Fall
$36°$	$61°$	$86°$	$65°$

Calculation:

The general form of the sinusoidal function is,

  $y=A\sin \left(kt+C\right)+h$

Amplitude is,

  $\begin{array}{l}A=\frac{86°-36°}{2}\\ A=25°\\ h=\frac{86°+36°}{2}\\ h=61°\\ \frac{2\pi }{k}=4\\ k=\frac{\pi }{2}\end{array}$

The general form is,

  $\begin{array}{l}y=25\sin \left(\frac{\pi }{2}t+C\right)+61\\ 36=25\sin \left(\frac{\pi }{2}t+C\right)+61\\ \sin \left(\frac{\pi }{2}t+C\right)=\frac{-25}{25}\\ C={\sin }^{-1}\left(\frac{\pi }{2}t+C\right)-\frac{\pi }{2}\\ C=90-90\\ C=-180\\ C=-\pi \end{array}$

Thus, general form of function is $y=25\sin \left(\frac{\pi }{2}t+C\right)+61$ .