Question 5. Consider the parametric equation x = 27 cos³ 0 and y = 27 sin³ 0.(a) Find the Cartesian equation of this parametric curve.(b) Find the arc length of this curve in the interval [0, 2π].

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Answered: Question 5. Consider the parametric…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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As the wheel of radius r cm in the figure rotates, the rod of length L attached to point P drives a piston back and forth in a straight line. Let x be the distance from the origin to point at the end of the rod as shown. (a) Use the Pythagorean Theorem to show that L² = (x − r cos 0)² + ² sin² 0. (b) Differentiate the equation in part (a) with respect to t to show that 0=2(x-r cos 0) (d+rsin 0df)+2r² sin cos de. dt (c) Calculate the speed of the piston when , assuming that r = 10 cm, L = 30 cm, and the wheel rotates at 4 revolutions per minute. L X = Piston moves back and forth e
1 Fig. 8 shows a sketch of part of the curve y = x sin 2x, where x is in radians. The curve crosses the x-axis at the point P. The tangent to the curve at P crosses the y-axis at Q. P y = x sin 2x Figh (i) Find Hence show that the x-coordinates of the turning points of the curve satisfy the equa dx tan 2x + 2x = 0). (ii) Find, in terms of 7, the x-coordinate of the point P. Show that the tangent PQ has equation 27x + 2y = ². Find the exact coordinates of Q. (iii) Show that the exact value of the area shaded in Fig. 8 is 7(x² − 2).

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Question 5. Consider the parametric equation x = 27 cos³0 and y = 27 sin³ 0. (a) Find the Cartesian equation of this parametric curve. (b) Find the arc length of this curve in the interval [0, 2π].