achieved by one full turn of the helix. Show that the length L of the helix sati

Question

achieved by one full turn of the helix. Show that the length L of the helix satisfies L2- ation vector is (hint: don't use e definitions of d also the prod- d that you have P2 + H2 6. There is a multistory parking ramp where e way out is a path in the shape of a he- lix that is wound around the outside of the building. As a car drives down this path at night its headlights shine a spot on the ground. Which curve is traced out by this light spot as the car drives all the way down? y had computed found at rigin

Explanation / Answer

There is a multistory parking ramp where the way out is a path in the shape of a helix that is wound around the outside of the building. As a car drives down this path at night its headlights shine a spot on the ground. Which curve is traced out by this light spot as the car drives all the way down?

Sol:

Say you're on the circle x2 + y2 = z2, starting at z = n, and every time unit, you drop down by 1 while going around an entire circle.

Then parametric equation = < rcos(2t) ,  rsin(2t) ,  n-t >

Here,

x(t) = rcos(2t), y(t) = rsin(2t), and z(t) = n-t.

Then dx/dt = -2r sin(2t), dy/dt = 2r cos(2t), and dz/dt = -1.

So, at time t, the headlights are pointed at

= < (x(t), y(t), z(t)) + p <dx/dt, dy/dt, dz/dt> > = <(rcos(2t),rsin(2t),n-t)+ p [-2r sin(2t), 2r cos(2t), -1 ] >

Assume the center of the parking ramp is the z-axis.

such that

z(t) + p (dz/dt) = 0

so n - t - p = 0, or p= n - t.

So, at time t, the headlights are pointed at

So (x(t), y(t), z(t)) + (n-t)<dx/dt, dy/dt, dz/dt>.

<(rcos(2t),rsin(2t),n-t)+ (n-t) [-2r sin(2t), 2r cos(2t), -1 ] >

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