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October 30th, 2009, 12:57 AM   #1
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Optimization Problem

Can you help me with the following optimization problem:
A minor sector is cut from a circular piece of paper with a radius of 6cm. The remaining (major) sector is used to form the curved surface of a cone. What size central angle should the remove sector have if the volume of the resulting cone is to be maximized?
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October 30th, 2009, 01:43 PM   #2
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Re: Optimization Problem

Suppose the base of the cone has radius r.

The distance from the tip to the edge of the base is 6 cm. By Pythagoras's Theorem, the height of the cone will be

So the volume of the cone, in terms of r, will be

Differentiate this and determine the values of r for which V ' (r) = 0. Pick the appropriate one.

Then find the angle by calculating the circumference of the cone's base, and dividing it by 12?, the circumference of the circle. This will give you the proportion of the whole circle which has been used to create the cone. You should find the amount that has been cut out to equal about 66.06°, and the cone's volume to be 87.06 cm².
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October 30th, 2009, 02:37 PM   #3
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Re: Optimization Problem

Since the circumference of the cone is 12? - 6?, where ? is in radians, r = (6? - 3?)/?.
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October 31st, 2009, 08:20 AM   #4
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That's circumference of the base of the cone, by the way.
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