practice6answers - PRACTICE6 Name Section 1 A ball rolls...

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PRACTICE6 Name Section 1. A ball rolls along a marked table and its position at any time can be determined by the parametric equations: x ( t ) = t 3 t 2 and y ( t ) = t 3 3 t . Determine dy dx when t = 3. dx dt = 3 t 2 2 t v v v v v t =3 = 21, and dy dt = 3 t 2 3 v v v v v t =3 = 24. dy dx v v v v v t =3 = dy/dt dx/dt v v v v v t =3 = 24 21 = 8 7 2. The paths v r 1 ( t ) = a t, t 2 A and v r 2 ( t ) = a sin ( t ) , sin (2 t ) A intersect when t = 0. De- termine the angle of intersection by determining the angle between their tangent vectors. a dx 1 dt , dy 1 dt Av v v v v t =0 = a 1 , 2 t A| t =0 = a 1 , 0 A a dx 2 dt , dy 2 dt Av v v v v t =0 = a cos ( t ) , 2 cos (2 t ) A| t =0 = a 1 , 2 A angle between curves: θ = arccos p a 1 , 0 A · a 1 , 2 A |a 1 , 0 A||a 1 , 2 A| P = arccos p 1 1 · 5 P 3. Determine the angle of intersection of the paths v r ( t ) = a t 2 + t + 2 , sin ( 3 t ) A and vs ( t ) = a 2 e 3 t , 2 t A as they cross at the time t = 0 through the point (2 , 0). a dx 1 dt , dy 1 dt Av v v v v t =0 = a 2 t + 1 , 3 cos ( 3 t ) A v v v t =0 = a 1 , 3 A a dx 2 dt , dy 2 dt Av v v v v t =0 = a 2 3 e 3 t
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