Use the lenghts A,B,R and d to find the coordinates of the point p. Thay is, Fin

Question

Use the lenghts A,B,R and d to find the coordinates of the point p. Thay is, Find equaltions for x and y in terms of these values.

335 Chapter 6 Group Project Herman Melville referred to the cycloid in this passage from Moby-Dick: DICK It was in the left-hand try-pot of the Pe circl quod, with the soapstone diligentt es gliding along a cycloid, my soapstone, for example, will assume the circle has radius r and is positioned on a rectangular co- ling round me, t in geometnhat I was first indirectly struck by the remarkable fact, that de scend from any point in precisely the same time To begin, we will ordinate system with its ce ure 2). Th ts center on the positive y-axis and tangent to the x-axis (Fig e x-axis y point P, ini Will serve as the line that the circle will roll along. We will choose e origin, to be the fixed point on the circumference of the circle e hcycloid. Figure 3 shows the position of the circle after it has rolled along the x-axis. We will use t to represent the angle (in radians) tially at th a short distance d a-axis. through which the circle has rotated. ERMAN MELVILLE Figure 2 Figure 3 1. If d is the distance that the circle has rolled, then what is the length of the arc s? 2. Use the lengths a, b, r, and d to find the coordinates of point P. That is, find equa- 3. Now use right triangle trigonometry to find a and b in terms of r and Use this to find a relationship between d and t tions for x and y in terms of these values 4. Using your answers to Questions 1 through 3, find equations for x and y in terms of r and t only. Because r is a constant, you now have parametric equations for the cycloid, using the angle t as the parameter! 5. Suppose that r = 1 . Complete the table by finding x and y for each value of t IS 2T. given. Then use your results to sketch the graph of the cycloid for O or to graph the cycloidforO$s6?. First, put your cal- Use your graphing calculat culator into radian and para 6. metric modes. Then define x and y using your equations from Question 4 (using r-1). Set your window so that 0 t 6?, 0 ss x 20 Melville referring to? What and 0 s y s 3 ch the cycloid. What property of the cycloid is 7. Resear ty does the cycloid have? What was the connection between tion at sea? Write a paragraph or two about your findings. Write the cycloid and navigation

Explanation / Answer

From the figure we can clearly see that ,

x-coordinate of P is : (d-a)

y-coordinate of P is : (r-b)

therfore,

P(x,y) = (d-a , r-b)

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