Chapter 9.6, Problem 20E

To determine

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

Answer to Problem 20E

  $y=-36.25\cos \left(\frac{\pi }{12}t\right)+34.25$ .

Explanation of Solution

Given information:

The highest point of bucket on the wheel is $70.5$ feet above the viewing platform and lowest point is $2$ feet below the viewing platform. The wheel makes complete turn every $24$ sec. The bucket at the edge of the wheel is at its lowest point when $t=0$ .

Calculation:

As per the given information −

The highest point of bucket on the wheel is $70.5$ feet above the viewing platform and lowest point is $2$ feet below the viewing platform. The wheel makes complete turn every $24$ sec. The bucket at the edge of the wheel is at its lowest point when $t=0$ .

Step 1: Find the maximum and minimum value.

From above graph, maximum value is $70.5$ and minimum value is $-2$ .

Step 2: Identify the vertical shift $k$ .

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

  $k=\frac{70.5+\left(-2\right)}{2}=34.25$ .

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.

  $\begin{array}{l}\left|a\right|=\frac{\text{Maximum value}-\text{Minimum value}}{2}=\frac{70.5-\left(-2\right)}{2}=36.25\\ \text{Since graph is a reflection, }a<0\\ \Rightarrow a=-36.25\end{array}$

Since the flywheel is rotating at the rate of $1$ revolution per $24$ seconds. So, period is $24$ sec.

  $\text{i}\text{.e}\text{. }\frac{2\pi }{b}=24\Rightarrow b=\frac{\pi }{12}$ .

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

  $y=-36.25\cos \left(\frac{\pi }{12}t\right)+34.25$

Hence, desired function of the given graph is $y=-36.25\cos \left(\frac{\pi }{12}t\right)+34.25$ .