Chapter 10.7, Problem 5MRPS

Solution

(a)

To find: Whether the graph be modelled in sin or cosine function. Give reasoning.

The value of $\sin C=\frac{3}{5},\cos C=\frac{4}{5}$ .

Given information:

The given graph is below:

  

The graph is to be modelled as sine function because midline of the graph intersects at $y=29$ .

(b)

To find: Give the trigonometric model for the graph.

The model for the given graph is . $y=27\sin \pi x+51$ .

Given information:

The given graph is:

  

The model for the graph is $y=a\sin b\left(t-h\right)+k$ .

First find amplitude:

  $\left|a\right|=\frac{M-m}{2}$

The maximum value $M$ from the graph is $73$ and minimum value is 29.

  $\begin{array}{l}\left|a\right|=\frac{73-29}{2}\\ \left|a\right|=\frac{54}{2}\\ \left|a\right|=27\end{array}$

Find the vertical shift $k$ apply the formula:

  $\begin{array}{l}k=\frac{M+m}{2}\\ k=\frac{73+29}{2}\\ k=\frac{102}{2}\\ k=51\end{array}$

The period of the function is 2.

  $\begin{array}{l}P=\frac{2\pi }{b}\\ 2=\frac{2\pi }{b}\\ b=\pi \end{array}$

Horizontal shift is $h=0$ .

Substitute all the values in $y=a\sin b\left(t-h\right)+k$ gives:

  $\begin{array}{l}y=27\sin \pi \left(t-0\right)+51\\ y=27\sin \pi t+51\end{array}$

(c)

To find: On which days of the year the average temperature in Coloredo is ${70}^0$ .

The number of days are $34.$

Given information:

The model for the given graph is . $y=27\sin \pi t+51$ .

Calculation:

Substitute $t=70$ in the given function gives:

  $\begin{array}{l}y=27\sin \pi \left(70\right)+51\\ y=27\left(-0.641\right)+51\\ y=-17.32+51\\ y=33.67\end{array}$

The number of days are $34.$