# MAIN CONE CONSTRUCTION

As a surface of revolution, the construction of this truncated cone is based on a general formula for finding the parametric equation of any surface of revolution. The measurements are taken from a side-elevation. Since the cone is not in its complete form, one needs the height of the complete cone in order to construct the existing shape. This is found by connecting various points along the center and sides of the truncated cone and extending them in order to estimate the position of the vertex. The estimated vertex is at 10.70 cm from the base of the cone and the radius of the cone at its base is 6.40 cm.

Notice that this radius is the same measured radius as the cylinder. This is because the initial construction of the model did not include a trapezoid layer or a base cone. There was only simply a cone atop a cylinder. This cone had the same measured radius as the cylinder; this was how the structure fit together. However, the inclusion of the other two pieces made for a more complex and accurate model. The main cone is the same as the original cone, yet it does not extend down to the cylinder. This cone begins at height 1.70 cm. Nevertheless, to construct this cone it is necessary to use its original measurements which include a base radius of 6.40 cm. The cone is truncated at a height of 7.49 cm.

Since the main cone is a surface of revolution, the parametric equation requires the radius to be expressed as a function of height. Let r represent radius and v represent height. Then, one sets up an equation of proportions

 6.4010.70 = r10.70 - v

10.70r = 6.40(10.70 - v)

 r = 68.48 - 6.40v10.70

r = 6.40 - .60v

As a result,

MainCone(u,v) = ((6.40 - .60v)cosu, (6.40 - .60v)sinu, v)

where u varies from 0 to 2pi and v from 1.70 to 7.49.

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