Chapter 7.3, Problem 53E

To determine

To find: the sinusoidal function that models the temperatures using given information.

Answer to Problem 53E

  $y=18\sin \left(\frac{\pi }{2}\left(t+2\right)\right)+68$

Explanation of Solution

Given information:

Use $t=1$ to represent winter. The table representing the average seasonal high temperature for North Carolina is shown below.

Winter	Spring	Summer	Fall
50 $°$	70 $°$	86 $°$	71 $°$

Calculation:

The general equation of a sine function for the given problem is as follows:

  $y=A\sin \left(\omega \left(t-C\right)\right)+B$

Here the amplitude is A, median value is B, angular velocity is $\omega$ and the phase shift is $C$ .

The amplitude can be evaluated as follows:

  $\begin{array}{c}A=\frac{\text{Highest temperature}-\text{Lowest temperature}}{2}\\ =\frac{86-50}{2}\\ =18\end{array}$

The median value can be evaluated as follows:

  $\begin{array}{c}B=A+\text{Lowest temperature}\\ =18+50\\ =68\end{array}$

The angular velocity can be evaluated as follows:

  $\begin{array}{c}\omega =\frac{2\pi }{n}\left[\text{Here }n\text{ is the total number of seasons, i}\text{.e}\text{.4}\right]\\ =\frac{2\pi }{4}\\ =\frac{\pi }{2}\end{array}$

The value of phase shift $C$ is 2.

Substitute all the values in equation (1), to get the sinusoidal function as shown below.

  $y=18\sin \left(\frac{\pi }{2}\left(t+2\right)\right)+68$