let vector r(t) = cos(pit)i + sin(pit)j + 2tk represent the position of a partic

Question

let vector r(t) = cos(pit)i + sin(pit)j + 2tk represent the position of a particle on a helix, where z is the height of the partice bove the ground.

a) is the particle ever moving downward? yes or no

b) when does the paritcle reach a pont 16 units above the ground?

t = ?

c) what is the velocity of the particle when it is 16 units above the ground? give an exact answer.

vector v = ? i + ? j + ? k

d) when the particle is 16 units above the ground, it leaves the helix and moves along the tangnt line. find parametric equations for this tangen line.

x(t) = ?

y(t) = ?

z(t) = ?

Explanation / Answer

1)no

2) We need 3t = 16 ==> t = 16/3.

3) v(t) = <-sin t, cos t, 3>
==> v(16/3) = <-sin(16/3), cos(16/3), 3>.

4) Since r(16/3) = <cos(16/3), sin(16/3), 16>, the equation of the tangent line at x = 16/3 (also using the result from part 2) yields

x = cos(16/3) - t sin(16/3), y = sin(16/3) + t cos(16/3), z = 16 + 3t.

I hope this helps!

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