V. Let C be the curve represented by R(t) = (cos 2t, 2√√3t, 5 — sin 2t).-π1. Determine the curvature of C at t =62. Reparametrize R using the arc length as parameter from the point P(-1,√√3, 5).3. Find the coordinates of point Q on C such that the directed arc length from P to Q is 7 units.

Answered: Image /qna-images/answer/0782b077-8f1c-4be1-a5f9-1c98d92b377b.jpg

Answered: V. Let C be the curve represented by…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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The curve C is parametrized by x = 2+2sint and y = 2cost, for 0 ≤ t ≤ π. Sketch C, noting the points P(0) when t = 0, and P(π) when t = π. Also note the direction for increasing values of t.
Suppose C is the curve given by r(t) = (3 cos(t), -3 sin(t), 4t). Show that both the curvature as well as the torsion of C are constant, and find their values.
Find T(t), N(t), and k(t) for the curve r(t) = (3 sint, 3 cos t, 4t). |3D Select one: 3 cos t, -sin t, 3. K = 25 а. Т — N = (-sin t, cos t, 0), 3. cos t, -sin t, N = (- sin t, cos t, 0), b. T = 25 4 3 cos t,-sin N = (- sin t, - cos t, 0), = 4 25 c. T = 3 d. T = cos t, sin t, N = (-sin t,- cos t, 0), 25 cost,sint,. 3 sin t, N = (-sin t, cos t, 0), e. Т- 25 45 45 94 35 35 35 35
Consider curve given by x=3et+3e-t and y=et-e-t find an equation for the curve without the parameter t. Hint: calculate x2 and y2
2. A particle travels in space a path described by on r(t)= (3", 4r",31), 0sisi. a) Give a rough sketch of the path, including points corresponding to t=0,1/2,1. b) How far does the particle travel along the path? c) Find the curvature of the path at t=1. What does the curvature indicate about the path at this time?
5. Sketch the curve r(t) = (you can plot it in Geogebra first and then transfer on to paper). Can you calculate the vectors T, B and N at t = 0 and at t 2πT? -
Graph the line that is described parametrically by (x, y) (2t,5 t), then.. %3D (a) Confirm that the point corresponding to t = O is exactly 5 units from (3,9). (b) Write a formula in terms of t for the distance from (3,9)to (2t, 5- t). (c) Find the other point on the line that is 5 units from (3,9). (d) Find the point on the line that minimizes the distance to (3,9). You may want a calculator.

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