   Chapter 10.3, Problem 59E

Chapter
Section
Textbook Problem

# Writing(a) Use a graphing utility to graph each set of parametric equations. x = t − sin t , y = 1 − cos t , 0 ≤ t ≤ 2 π x = 2 t − sin ( 2 t ) , y = 1 − cos ( 2 t ) , 0 ≤ t ≤ π (b) Compare the graphs of the two sets of parametric equations in part (a). When the curve represents the motion of a particle and t is time, what can you infer about the average speeds of the particle on the paths represented by the two sets of parametric equations?(c) Without graphing the curve, determine the time required for a particle to traverse the same path as in parts (a) and (b) when the path is modeled by x − 1 2 t − sin ( 1 2 t )   and   y = 1 − cos ( 1 2 t )

(a)

To determine

To-graph: The parametric equation x=tsint and y=1cost over the interval [0,2π] and the parametric equation x=2tsin(2t) and y=1cos(2t) over the interval [0,π] by the use of graphing utility.

Explanation

Given:

First set of parametric equations are x=tsint and y=1cost over the interval [0,2π].

And, the second set of parametric equations are x=2tsin(2t) and y=1cos(2t) over the interval [0,π].

Graph:

Consider, first the parametric equations, x=tsint and y=1cost over the interval [0,2π].

Now, graph the parametric equations by the use of Maple,

(b)

To determine

To calculate: The inference about the average speeds of the particle on the paths represented by the two set of parametric equations in part (a) when the curve represents motion and t represents time.

(c)

To determine

To calculate: The time required for a particle to traverse the same path as in parts (a) and (b) when the path is modelled by x=12tsin(12t) and y=1cos(12t).

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