Explanation Given: The parametric equation for the variable x is as below. x = cos 2 t The parametric equation for the variable y is as below. y = cos t The value t ranges from 0 to π 4 . Calculation: Substitute ( 0 ) for t in equation x = cos 2 t . x = cos 2 t = ( cos ( 0 ) ) 2 = 1 Substitute ( 0 ) for t in equation y = cos t . y = cos t = cos ( 0 ) = 1 t 0 π 4 π 3 5 12 π π 2 π 7 12 π 2 π 3 3 π 4 5 π 4 π x 1 0.499 0.249 0.066 0.00451 0.067 0.250 0.501 0.751 0.999 y 1 0.706 0.499 0.258 − 0.0007 − 0.259 − 0.500 − 0.707 0.866 − 0.999 The values of x and y for each step value of t is tabulated in the below table. Graph: Graph plotted for the parametric equation x = cos 2 t and y = cos t is shown below in Figure 1 . Refer the figure 1. The curve initiates from point ( 1 , 1 ) and returns to the point ( 1 , 1 ) between the range 0 to π , and the particle travels four times for the values of parameter t ( 0 , 2 π , 4 π ) . The distance traveled by the particle will be 4 times of length L . The total distance traveled by the particle is d = 4 × L Write the length formula for the curve. L = ∫ α β ( d x d t ) 2 + ( d y d t ) 2 d t Differentiate the variable x with respect to t . x = cos 2 t d x d t = − 2 cos t sin t Differentiate the variable y with respect to t . y = cos t d y d t = − sin t Write the length of the curve formula as below. L = ∫ α β ( d x d t ) 2 + ( d y d t ) 2 d t Substitute ( − 2 cos t sin t ) for d x d t and ( − sin t ) for d y d t in the above equation. L = ∫ α β ( d x d t ) 2 + ( d y d t ) 2 d t = ∫ 0 π ( − 2 cos t sin t ) 2 + ( − sin t ) 2 d t = ∫ 0 π 4 c o s 2 t sin 2 t + sin 2 t d t = ∫ 0 π sin t 4 c o s 2 t + 1 d t Substitute variable u for cos t . u = cos t sin t d u = d t Substitute ( 0 ) for t in equation u = cos t

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Chapter 10.2, Problem 52E

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To find: the distance traveled by a particle for the parametric equation x=cos2t and y=cost .

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Chapter 10.2, Problem 51E

Chapter 10.2, Problem 53E

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Ch. 10.1 - Sketch the curve by using the parametric equations...Ch. 10.1 - Sketch the curve by using the parametric equations...Ch. 10.1 - Prob. 3E Ch. 10.1 - Prob. 4E Ch. 10.1 - Prob. 5E Ch. 10.1 - Prob. 6E Ch. 10.1 - Prob. 7E Ch. 10.1 - Prob. 8E Ch. 10.1 - Prob. 9E Ch. 10.1 - Prob. 10E

Ch. 10.1 - Prob. 11E Ch. 10.1 - Prob. 12E Ch. 10.1 - Prob. 13E Ch. 10.1 - Prob. 14E Ch. 10.1 - Prob. 15E Ch. 10.1 - Prob. 16E Ch. 10.1 - Prob. 17E Ch. 10.1 - Prob. 18E Ch. 10.1 - Prob. 19E Ch. 10.1 - Prob. 20E Ch. 10.1 - Describe the motion of a particle with position...Ch. 10.1 - Prob. 22E Ch. 10.1 - Prob. 23E Ch. 10.1 - Prob. 24E Ch. 10.1 - Prob. 25E Ch. 10.1 - Prob. 26E Ch. 10.1 - Prob. 27E Ch. 10.1 - Match the parametric equations with the graphs...Ch. 10.1 - Prob. 29E Ch. 10.1 - Prob. 30E Ch. 10.1 - Prob. 31E Ch. 10.1 - Use a graphing device and the result of Exercise...Ch. 10.1 - Prob. 33E Ch. 10.1 - Prob. 34E Ch. 10.1 - Prob. 35E Ch. 10.1 - Prob. 36E Ch. 10.1 - Prob. 37E Ch. 10.1 - Prob. 38E Ch. 10.1 - Prob. 39E Ch. 10.1 - Prob. 40E Ch. 10.1 - Prob. 41E Ch. 10.1 - Prob. 42E Ch. 10.1 - Prob. 43E Ch. 10.1 - (a) Find parametric equations for the set of all...Ch. 10.1 - Suppose that the position of one particle at time...Ch. 10.1 - Prob. 46E Ch. 10.1 - Prob. 47E Ch. 10.1 - Prob. 48E Ch. 10.1 - Prob. 49E Ch. 10.1 - Prob. 50E Ch. 10.1 - Prob. 51E Ch. 10.1 - Prob. 52E Ch. 10.2 - Prob. 1E Ch. 10.2 - Prob. 2E Ch. 10.2 - Prob. 3E Ch. 10.2 - Prob. 4E Ch. 10.2 - Prob. 5E Ch. 10.2 - Prob. 6E Ch. 10.2 - Prob. 7E Ch. 10.2 - Prob. 8E Ch. 10.2 - Prob. 9E Ch. 10.2 - Find an equation of the tangent to the curve at...Ch. 10.2 - Prob. 11E Ch. 10.2 - Prob. 12E Ch. 10.2 - Prob. 13E Ch. 10.2 - Prob. 14E Ch. 10.2 - Find dy/dx and d2y/dx2. 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Prob. 8E Ch. 10.4 - Prob. 9E Ch. 10.4 - Prob. 10E Ch. 10.4 - Prob. 11E Ch. 10.4 - Prob. 12E Ch. 10.4 - Prob. 13E Ch. 10.4 - Prob. 14E Ch. 10.4 - Prob. 15E Ch. 10.4 - Prob. 16E Ch. 10.4 - Prob. 17E Ch. 10.4 - Prob. 18E Ch. 10.4 - Prob. 19E Ch. 10.4 - Prob. 20E Ch. 10.4 - Prob. 21E Ch. 10.4 - Prob. 22E Ch. 10.4 - Prob. 23E Ch. 10.4 - Prob. 24E Ch. 10.4 - Find the area of the region that lies inside the...Ch. 10.4 - Prob. 26E Ch. 10.4 - Prob. 27E Ch. 10.4 - Prob. 28E Ch. 10.4 - Prob. 29E Ch. 10.4 - Prob. 30E Ch. 10.4 - Prob. 31E Ch. 10.4 - Prob. 32E Ch. 10.4 - Prob. 33E Ch. 10.4 - Prob. 34E Ch. 10.4 - Prob. 35E Ch. 10.4 - Prob. 36E Ch. 10.4 - Prob. 37E Ch. 10.4 - Prob. 38E Ch. 10.4 - Prob. 39E Ch. 10.4 - Prob. 40E Ch. 10.4 - Prob. 41E Ch. 10.4 - Prob. 42E Ch. 10.4 - The points of intersection of the cardioid r = 1 +...Ch. 10.4 - When recording live performances, sound engineers...Ch. 10.4 - Prob. 45E Ch. 10.4 - Find the exact length of the polar curve. 46. r =...Ch. 10.4 - Prob. 47E Ch. 10.4 - Prob. 48E Ch. 10.4 - Find the exact length of the curve. Use a graph to...Ch. 10.4 - Find the exact length of the curve. 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the distance traveled by a particle for the parametric equation x = cos 2 t and y = cos t .

Solution Summary: The author explains the parametric equation x=mathrmcos2t, and the corresponding equation for the variable

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To find: the distance traveled by a particle for the parametric equation x=cos2t and y=cost .

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