A Christmas tree has the shape of the conical helix. Find the length of the Christmas tree. A Christmas tree has the shape of the conical helix. The helix has the circular base of 1 foot diameter, 
and it rises three complete turns. Find the length of the Christmas tree.
Image the goes with the problem
Please explain the steps. Thanks!
 A: Here's a start: a (cylindrical) helix could be written as $x(t) = \cos t$, $y(t) = \sin t$, $z(t) = t$.  You should check that this helix actually lies on the cylinder $x^2 + y^2 = 1$.  We want to modify this so that the helix instead lies on the cone $(1-z)^2 = x^2 + y^2$.  See here, your "tree" should lie on the portion of this cone between $z=0$ and $z=1$.
We can modify so that the helix lies on the cone: $x(t) = (1-t)\cos t$, $y(t) = (1-t)\sin t$, $z(t) = t$, where $t$ ranges from $0$ to $1$.  You should check that this actually lies on the cone.  However, it does not make three rotations from $0$ to $1$.  We can adjust this by adjusting the arguments inside cosine and sine, since these affect the speed of the rotation:
$$x(t) = (1-t)\cos(6\pi t),$$
$$y(t) = (1-t)\sin(6\pi t),$$
$$z(t) = t,$$
$$ 0 \leq t \leq 1.$$
You should make sure you understand why this lies on the cone, and why it makes three rotations from $t=0$ to $t=1$.  (See here to visualize).
Now all that's left is to compute the arc length of this curve, which you can do with the formula:
$$\int_0^1 \sqrt{x'(t)^2 + y'(t)^2 + z'(t)^2} \ dt.$$
