Chapter 9.6, Problem 19E

To determine

To find: A function model for height $h$ (in feet) of the handle as function of time $t$ .

Answer to Problem 19E

  $y=-2.5\cos \left(\pi t\right)+6.5$ .

Explanation of Solution

Given information:

Flywheel of sewing machine is $5$ feet in diameter. The highest point is $9$ feet above the ground and lowest point is $4$ feet above the ground. The wheel makes complete turn every $2$ sec. The handle at the edge of the flywheel is at its lowest point when $t=0$ .

Calculation:

As per the given information −

Flywheel of sewing machine is $5$ feet in diameter. The highest point is $9$ feet above the ground and lowest point is $4$ feet above the ground. The wheel makes complete turn every $2$ sec. The handle at the edge of the flywheel is at its lowest point when $t=0$ .

Step 1: Find the maximum and minimum value.

From above graph, maximum value is $9$ and minimum value is $4$ .

Step 2: Identify the vertical shift $k$ .

The value of $k$ is the mean of maximum and minimum value, i.e.

  $k=\frac{9+4}{2}=6.5$

Step 3: Decide whether graph should be modeled as sine or cosine function.

When $t=0$ , height is at its minimum. So, use cosine function whose graph is a reflection in $x-$ axis with no horizontal shift .

Step 4: Find the amplitude and period.

  $\begin{array}{l}\left|a\right|=\frac{\text{Maximum value}-\text{Minimum value}}{2}=\frac{9-4}{2}=2.5\\ \text{Since graph is a reflection, }a<0\\ \Rightarrow a=2.5\end{array}$

Since the flywheel is rotating at the rate of $1$ revolution per two seconds. So, period is $2$ sec.

  $\text{i}\text{.e}\text{. }\frac{2\pi }{b}=2\Rightarrow b=\pi$ .

Step $5$ : The function graph would be given as −

  $y=-2.5\cos \left(\pi t\right)+6.5$

Hence, desired function of the given graph is $y=-2.5\cos \left(\pi t\right)+6.5$ .