2.1 Traversing Circles

Introduction

Circular Functions

Properties of Circular Functions

In the context of the ferris wheel pictured in Figure 2.1.1 in the text, assume that the height, $h$, of the moving point (the cab in which you are riding), and the distance, $d$, that the point has traveled around the circumference of the ferris wheel are both measured in meters.

Further, assume that the circumference of the ferris wheel is $150$ meters. In addition, suppose that after getting in your cab at the lowest point on the wheel, you traverse the full circle several times.

Recall that the circumference, $C$, of a circle is connected to the circle's radius, $r$, by the formula $C = 2\pi r$. What is the radius of the ferris wheel? How high is the highest point on the ferris wheel?

How far along the circle is the point $P_1$ from $P_0$? Why?

Label the subsequent points in the figure $P_2$, $P_3$, $\ldots$ as we move counterclockwise around the circle. What is the exact $y$-coordinate of the point $P_2$? of $P_4$? Why?

Determine the $y$-coordinates of the remaining points on the circle (exactly where possible, otherwise approximately) and hence complete the entries in the following table that track the height, $h$, of the point traversing the circle as a function of distance traveled, $d$. Note that the $d$-values in the table correspond to the point traversing the circle more than once.

$d$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$ | $13$ | $14$ | $15$ | $16$

$h$ | $2$ |

|

|

|

|

|

|

|

By plotting the points in the table in part (c) and connecting them in an intuitive way, sketch a graph of $h$ as a function of $d$ over the interval $0 \le d \le 16$ on the axes provided in the figure in part (a). Clearly label the scale of your axes and the coordinates of several important points on the curve.

What is similar about your graph in comparison to the one in Figure 2.1.5 in the text? What is different?

What will be the value of $h$ when $d = 51$? How about when $d = 102$?

Determine the period $p$, midline $y = m$, and amplitude $a$ of the function $f$.

What is the greatest distance the weight is displaced from the wall? What is the least distance the weight is displaced from the wall? What is the range of $f$?

Determine the average rate of change of $f$ on the intervals $[4,4.25]$ and $[4.75,5]$. Write one careful sentence to explain the meaning of each (including units). In addition, write a sentence to compare the two different values you find and what they together say about the motion of the weight.

Based on the periodicity of the function, what is the value of $f(6.75)$? of $f(11.25)$?