Decorating a Tree with a Ribbon

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This Demonstration calculates the length of a ribbon wrapped around a tapered post or a tree (geometrically, a frustum, which is a truncated cone) in two ways:

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1. approximately, by unfolding the cone and adding the lengths of all segments, where is the number of turns.

2. exactly, using the formula for arc length, , where is the height of frustum and derivatives are taken with respect to , the height, of the components of the parametric equation of the ribbon (as a curve):

,

,

,

where is the radius of the frustum at height and slope .

In addition, the Demonstration calculates the surface area and the volume for different heights and slopes.

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Contributed by: Abraham Gadalla (July 2015)
Open content licensed under CC BY-NC-SA


Snapshots


Details

The result of revolving the line about the axis is a cone. Here is the slope and is the intercept of the line. In order to keep the base radius of the frustum constant at 3 feet while changing the slope, the intercept value has to be a function of the slope.

By unfolding the cone, the subtended angle is calculated by the simple formula:

.

Here is the arc length of the base of the cone and is its the side length.

The length of one turn of the ribbon is the arc length of the ribbon on the unfolded frustum. It is approximated by the length of the segment on the unfolded frustum.

The length of the side of the frustum is .

The distance between two consecutive turns is .

The lateral surface area of the frustum is , where is the radius at the top of the frustum.

Finally, the volume of the frustum is .



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