Chapter 10.5, Problem 3GP

Solution

To state: The change in the model in example 2 if the lowest point of a rope is 5 inches above the ground and the highest point is 70 inches above the ground.

The resultant values are $h=-32.5\cos 4\pi t+37.5$ .

Given information:

From example 2, the old model is $h=-36\cos 4\pi t+39$ . It is given that the new rope’s maximum height is 70 inches and minimum height is 5 inches.

Formula used: The amplitude formula is $\text{Amplitude}\left(a\right)=\frac{M-m}{2}$ .

Calculation:

Consider the old model: $h=-36\cos 4\pi t+39$

This function is of the form of $y=a\mathrm{cosb}\left(x-h\right)+k$ . Then the values are:

  $a=-36,b=4\pi ,h=0,\text{ and }k=39$ .

The given values of $M\text{ is 70 and }m\text{ is 5}$ .

So the Amplitude is:

  $\begin{array}{c}\text{Amplitude}\left(a\right)=\frac{M-m}{2}\\ =\frac{70-5}{2}\\ =32.5\end{array}$

Since the graph is a reflection, the value of $a<0$ . Thus, the value $a\text{ will be }-32.5$ .

The vertical shift is:

  $\begin{array}{c}\text{Vertical shift}\left(k\right)=\frac{M+m}{2}\\ =\frac{70+5}{2}\\ =37.5\end{array}$

Now model the new function by substituting the values $a=-32.5,b=4\pi ,h=0,k=37.5,y=h$ .

The function will become:

  $\begin{array}{c}h=-32.5\cos 4\pi \left(t-0\right)+37.5\\ =-32.5\cos 4\pi t+37.5\end{array}$

Therefore, the resultant sinusoid is $h=-32.5\cos 4\pi t+37.5$ .