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!
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$\begingroup$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: $$\int_0^1 \sqrt{x'(t)^2 + y'(t)^2 + z'(t)^2} \ dt.$$
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