Activity 2.1.2.
Consider the circle pictured in the following figure that is centered at the point \((2,2)\) and that has circumference \(8\text{.}\) Assume that we track the \(y\)-coordinate (that is, the height, \(h\)) of a point that is traversing the circle counterclockwise and that it starts at \(P_0\) as pictured.
(a)
(b)
Label the subsequent points in the figure \(P_2\text{,}\) \(P_3\text{,}\) \(\ldots\) as we move counterclockwise around the circle. What is the exact \(y\)-coordinate of the point \(P_2\text{?}\) of \(P_4\text{?}\) Why?
(c)
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, \(h\text{,}\) of the point traversing the circle as a function of distance traveled, \(d\text{.}\) Note that the \(d\)-values in the table correspond to the point traversing the circle more than once.
| \(d\) | \(0\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) | \(13\) | \(14\) | \(15\) | \(16\) |
| \(h\) | \(2\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) | \(\) |
(d)
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.
(e)
What is similar about your graph in comparison to the one in Figure 2.1.5 in the text? What is different?
