Chapter 0.6, Problem 43E

To determine

To calculate: The value of $A$ , $B$ , and $C$ with the help of the given graph. of the matrix $C$ by using the first row.

Answer to Problem 43E

The value of determinant is 3.

Explanation of Solution

Given Information:

The graph is,

  

Formula Used:

The period is defined as,

  $P=\left|\frac{2\pi }{B}\right|$

The general function is defined as,

  $y=A\cos \left(Bx\right)+C$

Where, $A$ is amplitude, $C$ is vertical shift.

Calculation:

Consider the given information,

It is given that the temperature range from a $40.5°F$ in January to a high of $80°F$ . So find the amplitude.

  $\begin{array}{c}A=\left|\frac{80-40.5}{2}\right|\\ =\left|\frac{39.5}{2}\right|\\ =19.75\end{array}$

As the rotation of the graph of the function in opposite side, so the amplitude will be negative.

Now, find the value of $B$ .

As the total period covered by the graph is 12 points. So $P=12$ .

  $\begin{array}{c}12=\frac{2\pi }{B}\\ B=\frac{2\pi }{12}\\ =\frac{\pi }{6}\end{array}$

Now, find the value the vertical displacement $C$ .

As the initial point of the displacement is $19.75$ . Now, the maximum point on the given graph is 80.

  $\begin{array}{c}C=80-19.75\\ =60.25\end{array}$

Substitute all the values in the formula.

  $y=-19.75\cos \left(\frac{\pi }{6}x\right)+60.25$

Therefore, the obtained function is $y=-19.75\cos \left(\frac{\pi }{6}x\right)+60.25$ .