Problem: Let r(t) = cos(t) i + sin(t) j + 4t k represents the position of a particle on a helix, where z is the height of the particle above the ground.

(d) When it is 28 units above the ground, the particle leaves the helix and moves along the tangent line. Find parametric equations for this tangent line.

My Answer So Far:

I know that the particle is 28 units above the ground at t=7, and I know that the vector for velocity evaluated at t=7 is $\displaystyle v= -\pi sin(7\pi)i + \pi cos(7\pi)j + 4k$.

Therefore, the parametric equations should be found using:

p(t) = r(5) + (t-7)(v(5))

When I plug everything in and solve, I get the following parametric equations:

$\displaystyle x(t) = cos(7\pi)-t\pisin(7\pi)$
$\displaystyle y(t) = sin(7pi)+t\picos(7pi)-7picos(7pi)$
$\displaystyle z(t) = 28+4(t-7)$

The first and last one are correct on my online homework, but the second one (the y(t) parametric equation) is wrong somehow. I cannot figure out why it's wrong. I've redone it numerous times and get the same answer!

Any help would be much appreciated!