Question

Exploration 16.2:A Parameterization of Movement in the Plane Since you are approximating the various chosen s lengths, it might be helpful to note that 1/8 of an inch is equal to 0.125 inches and thus 1/16 of an inch is approximately equal to Often it is useful to investigate a rule or relation between ordered pairs in the plane as they relate to yet a third variable or parameter. In this investigation, movement of an object in the plane is described by parametric equations. Using a non-standard approach, we will describe the path of an object as it moves around the circumference of a circle. The relationship between circumference arc length s traversed by the object and associated values we will call w(s) and h(s) will be described. Using the origin as our reference point, the value of w(s) will represent the horizontal distance in the plane and h(s) will represent the vertical distance in the plane used to locate the endpoint of the path of our object as it moves in a counter-clockwise direction along the circumference of a unit circle. The assumption will be that the chosen portion of the circumference is measured from the point (1, 0) In addition to recording the chosen circumference s lengths, one needs to also record the ws) and h(s) distances associated with each s length as pictured in Figure 1. Recall that wis) represents the horizontal distance in the plane and h(s) represents the vertical distance in the plane used to locate the endpoint of an arbitrarily chosen length s along the circumference of your circle. It will be helpful to create a table of values in the format of Table 1. Keep in mind that you must denote the positive or negative direction of your recorded ws) and h(s) segment lengths depending on which quadrant you are working in when recording values. It is suggested that you take ten to twelve measurements as you progress around the circumference of the cirele. Table 1 To start, a circle of radius approximately one inch (for simplicity) centered at the origin of your coordinate system has been constructed in Figure 1. A thin piece of string of approximately 13 inches in length will be needed for this exploration. INOTE: The scale on the supplied ruler and "one inch" unit circle can be taken as the same. Please keep in mind that, due to paper copying irregularities, the provided ruler is likely not an accurate representation of a standard inch measurement. 1. Describe any patterns you notice in the relationship between s and w(s) and between s and h(s) respectively 2. Next, use the data recorded in Table 1 to construct two different graphs. The first graph should be a graph of w(s) values as a function of the associated s values, and the second graph should be of h(s) values as a function of the associated s values. The graphs of Figures 2 and 3 are provided for your use. Be sure to label and scale each graph. 1* Figure 1. (Not drawn to scale) h(s) Exploration procedure: Place one end of a string at the point (1,0) and run a length of the string along the circumference of the provided circle. Take approximate measurements of your s lengths at rational multiples of 2r proceeding around the circle in a counter-clockwise fashion Record readings in this manner starting from zero to twice around the circle. Measure your s string lengths using the provided ruler of Figure 2 12345678910111213น15 3. Fully describe the pattern of each of the relationships now that you have graphed them. 4. Can you find the value for s such that w(s) and h(s) are of equal value? 5. Can you find the value for s such that w(s) is half that of s? Also, what is the value of Figure 2. I[Ruler: use inches to measure string h(s) at this point?

Answer 1

rive C, bes tane value t wes eand

It is always advised to ask 1 question at a time for better result.

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