Suppose r⃗ (t)=cos(πt)i+sin(πt)j+5tkr→(t)=cos( πt)i+sin(πt)j+5tk 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 2020? t=t= (b) What is the velocity of the particle when its height is 2020? v⃗ =v→= (c) When the particle has height 2020, 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.

a) t = 4

b) v = pi j + 5 k

c) rt = 1i + (pi t) j + (20 +5t )k

Step-by-step explanation:

You have the following vector equation for the position of a particle:

   (1)

(a) The height of the helix is given by the value of the third component of the position vector r, that is, the z-component.

For a height of 20 you have:

(b) The velocity of the particle is the derivative, in time, of the vector position:

   (2)

and for t=4 (height = 20):

(c) The vector parametric equation of the tangent line is given by:

     (3)

ro: position of the particle for t=4

Then you replace ro and v in the equation (3):

y - y1 = m(x- x1)

y - (-5) = 2/3(x -0)

y + 5 = 2/3x

y = 2/3x - 5

answer: 38.47 is your answer

step-by-step explanation: hope i you and who ever sees this answer have a great rest of your day : )