Answer Happy

Cycloids The path of a fixed point on a circle rolling along a straight line is given by the parametric equations x = r(8-sin(8)), y=r(1- cos(8)) where r is the radius of the circle and the parameter is the angle of rotation. Drag the r slider handle, click on the slider bar, or click on a number above the slider to select the radius for the circle. Click the Animate button to start the circle rolling and observe the path formed by the fixed point on the circle. You can also use the slider handle to increase or decrease 8 gradually and the 8 slider to adjust the number of rotations. max
160 in 5 4 3- 2 1 0- -1 Pº 0 Cycloids -4 -3 -2 -1 0 0.5 0 1x -m 1.0 1.5 2.0 2.5 3.0 X 1 2 3 4567 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 2.00 Omax 6π ● x=r(0 - sin(0)) y = r(1-cos()) 2x 3x 2x 4x 5 Animate ▶ 5x 8x 44 6 6.28
Select "Cycloids from the pull-down menu. (a) With the default value r=1, graph the parametric curve for 0 s8 s 8r. How many arches do you see on the graph for this domain? What interval of 8 is needed to draw one arch? (Enter your answer using interval notation.) [0, 2π] What is the width of one arch? 2π (b) Set r= 2 and graph the cycloid. What interval of 8 is needed to draw one arch? (Enter your answer using interval notation.) [0, 2π] What is the width of one arch? 4元 Set r = 3. What interval of 9 is needed to draw one arch? (Enter your answer using interval notation.) [0, 2x] What is the width of one arch? 6π (c) What is the width of one arch in terms of r? What about the height? 27 Can you explain why these results are to be expected?