Suppose (t) = cos(pi t)i + sin(pit)j + 5tk represents the position of a particle

Question

Suppose (t) = cos(pi t)i + sin(pit)j + 5tk represents the position of a particle on a helix, where z is the height of the particle. (a) What is t when the particle has height 20? t (b) What is the velocity of the particle when its height is 20? V = (e) When the particle has height 20, it leaves the helix and moves along the tangent line at the constant velocity found in part (b). Find a vector parametric equation for the position of the particle (in terms of the original parameter t) as it moves along this tangent line. L(t)=

Explanation / Answer

1) We need 5t = 20 ==> t = 4.

2) v(t) = <-sin t, cos t, 3>
==> v(4= <-sin(4), cos(4), 3>.

3)

Since r(4) = <cos(4), sin(4), 20>, the equation of the tangent line at x = 4 (also using the result from part 2) yields

x = cos(4) - t sin(4), y = sin(4) + t cos(4), z = 20 + 5t.

I hope this helps!

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