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Unformatted text preview: rhodes (ajr2283) – Exam02Review – Gilbert – (56380) 1 This printout should have 19 questions. Multiplechoice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Describe the motion of a particle with posi tion P ( x, y ) when x = 4 sin t , y = 3 cos t as t varies in the interval 0 ≤ t ≤ 2 π . 1. Moves once counterclockwise along the ellipse (4 x ) 2 + (3 y ) 2 = 1 , starting and ending at (0 , 3). 2. Moves along the line x 4 + y 3 = 1 , starting at (4 , 0) and ending at (0 , 3). 3. Moves once clockwise along the ellipse (4 x ) 2 + (3 y ) 2 = 1 , starting and ending at (0 , 3). 4. Moves once counterclockwise along the ellipse x 2 16 + y 2 9 = 1 , starting and ending at (0 , 3). 5. Moves along the line x 4 + y 3 = 1 , starting at (0 , 3) and ending at (4 , 0). 6. Moves once clockwise along the ellipse x 2 16 + y 2 9 = 1 , starting and ending at (0 , 3). correct Explanation: Since cos 2 t + sin 2 t = 1 for all t , the particle travels along the curve given in Cartesian form by x 2 16 + y 2 9 = 1 ; this is an ellipse centered at the origin. At t = 0, the particle is at (4 sin0 , 3 cos0), i.e. , at the point (0 , 3) on the ellipse. Now as t increases from t = 0 to t = π/ 2, x ( t ) increases from x = 0 to x = 4, while y ( t ) decreases from y = 3 to y = 0 ; in particular, the particle moves from a point on the positive yaxis to a point on the positive xaxis, so it is moving clockwise . In the same way, we see that as t increases from π/ 2 to π , the particle moves to a point on the negative yaxis, then to a point on the negative xaxis as t increases from π to 3 π/ 2, until finally it returns to its starting point on the positive yaxis as t increases from 3 π/ 2 to 2 π . Consequently, the particle moves clockwise once around the ellipse x 2 16 + y 2 9 = 1 , starting and ending at (0 , 3). keywords: motion on curve, ellipse 002 10.0 points A ladder 13 feet in length slides down a wall as its bottom is pulled away from the wall as shown in rhodes (ajr2283) – Exam02Review – Gilbert – (56380) 2 13 x y θ P Using the angle θ as parameter, find the parametric equations for the path followed by the point P located 3 feet from the top of the ladder. 1. (3 sec θ, 10 tan θ ) 2. (10cos θ, 3 sin θ ) 3. (10tan θ, 3 sec θ ) 4. (3 cos θ, 10sin θ ) correct 5. (3 sin θ, 10cos θ ) 6. (10sin θ, 3 cos θ ) 7. (10sec θ, 3 tan θ ) 8. (3 tan θ, 10 sec θ ) Explanation: By right triangle trigonometry, the coordi nates ( x, y ) are given respectively by x = 3 cos θ , y = (13 − 3) sin θ . Consequently, the curve traced out by P has the parametric form (3 cos θ, 10 sin θ ) for 0 ≤ θ ≤ π/ 2. Eliminating θ , we see that P traces out the portion of the ellipse x 2 9 + y 2 100 = 1 in the first quadrant....
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This note was uploaded on 09/27/2010 for the course M 408 D taught by Professor Textbookanswers during the Spring '07 term at University of Texas at Austin.
 Spring '07
 TextbookAnswers
 Multivariable Calculus

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