Gilbert_ReviewExam02sol

# Gilbert_ReviewExam02sol - moseley(cmm3869 – Exam02Review...

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Unformatted text preview: moseley (cmm3869) – Exam02Review – Gilbert – (56380) 1 This print-out should have 19 questions. Multiple-choice 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 = 3 sin t , y = 2 cos t as t varies in the interval 0 ≤ t ≤ 2 π . 1. Moves along the line x 3 + y 2 = 1 , starting at (0 , 2) and ending at (3 , 0). 2. Moves once clockwise along the ellipse x 2 9 + y 2 4 = 1 , starting and ending at (0 , 2). correct 3. Moves once counterclockwise along the ellipse (3 x ) 2 + (2 y ) 2 = 1 , starting and ending at (0 , 2). 4. Moves along the line x 3 + y 2 = 1 , starting at (3 , 0) and ending at (0 , 2). 5. Moves once counterclockwise along the ellipse x 2 9 + y 2 4 = 1 , starting and ending at (0 , 2). 6. Moves once clockwise along the ellipse (3 x ) 2 + (2 y ) 2 = 1 , starting and ending at (0 , 2). 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 9 + y 2 4 = 1 ; this is an ellipse centered at the origin. At t = 0, the particle is at (3 sin0 , 2 cos0), i.e. , at the point (0 , 2) on the ellipse. Now as t increases from t = 0 to t = π/ 2, x ( t ) increases from x = 0 to x = 3, while y ( t ) decreases from y = 2 to y = 0 ; in particular, the particle moves from a point on the positive y-axis to a point on the positive x-axis, 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 y-axis, then to a point on the negative x-axis as t increases from π to 3 π/ 2, until finally it returns to its starting point on the positive y-axis as t increases from 3 π/ 2 to 2 π . Consequently, the particle moves clockwise once around the ellipse x 2 9 + y 2 4 = 1 , starting and ending at (0 , 2). keywords: motion on curve, ellipse 002 10.0 points A ladder 16 feet in length slides down a wall as its bottom is pulled away from the wall as shown in moseley (cmm3869) – Exam02Review – Gilbert – (56380) 2 16 x y θ P Using the angle θ as parameter, find the parametric equations for the path followed by the point P located 4 feet from the top of the ladder. 1. (4 sec θ, 12 tan θ ) 2. (12sin θ, 4 cos θ ) 3. (12sec θ, 4 tan θ ) 4. (12cos θ, 4 sin θ ) 5. (12tan θ, 4 sec θ ) 6. (4 sin θ, 12cos θ ) 7. (4 tan θ, 12 sec θ ) 8. (4 cos θ, 12sin θ ) correct Explanation: By right triangle trigonometry, the coordi- nates ( x, y ) are given respectively by x = 4 cos θ , y = (16 − 4) sin θ . Consequently, the curve traced out by P has the parametric form (4 cos θ, 12 sin θ ) for 0 ≤ θ ≤ π/ 2. Eliminating θ , we see that P traces out the portion of the ellipse x 2 16 + y 2 144 = 1 in the first quadrant....
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Gilbert_ReviewExam02sol - moseley(cmm3869 – Exam02Review...

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