Need help with Calc 3 prob.: A Christmas tree has the shape of the conical helix....

[z400jt]

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.

I have this problem for my next Calculus test and I'm pretty stumped on it. Could anyone explain how to do it? Thanks.

 

[z400jt]

Help with Calc 3 Review Problem

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.

 

[stapel]

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.

I have this problem for my next Calculus test and I'm pretty stumped on it. Could anyone explain how to do it? Thanks.

They gave you a fairly lengthy explanation here, showing how to parametrize the curve:

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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 x² + y² = 1. We want to modify this so that the helix instead lies on the cone (1 − z)² = x² + y². 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(6πt),

. . .y(t) = (1 − t) sin(6πt),

. . .z(t) = t, for

. . .0 ≤ t ≤ 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:

. . .\(\displaystyle \displaystyle \int_0^1 \, \sqrt{\strut 10\, x'(t)^2\, +\, y'(t)^2\, +\, z'(t)^2\, }\, dt\)

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Where are you stuck in applying that information? What have you used to supplement your study (such as this or this)?

Please be complete. Thank you!