Suppose r (t)=cos(t) i +sin(t) j +4t k r(t)=cos(t)i+sin(t)j+4tk represents the p

Question

Suppose r (t)=cos(t)i+sin(t)j+4tkr(t)=cos(t)i+sin(t)j+4tk represents the position of a particle on a helix, where zz is the height of the particle.

(a) What is tt when the particle has height 88?


(b) What is the velocity of the particle when its height is 88?


(c) When the particle has height 88, 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 tt) as it moves along this tangent line.
L(t)=

Explanation / Answer

Given r(t)=<cos((t),sin(t),4t>

(a) Since the variable z represents the height.

So, we need 4t = 88     => t = 22

(b) v(t) = r'(t) = <- sin(t), cos(t), 4>.

=> v(22) = <0, , 4>.

(c) Since r(t)=<cos((t),sin(t),4t> => r(22)=<1,0,88>

and r'(4) = <0, , 4>,

so, the tangent line has equation

L(t) = <1, 0, 88> + t<0, , 4> = <1, t, 4t + 88>

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