A sector of a circle with central angle theta is cut from a circle with radius R. The edges of teh sector are then glued to form a cone. Find the value of theta that produces a cone with maximum volume. Please show all work!

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Lets look at this for a moment together ..from a top view down on the circle...theta could be 1 deg or 359 deg .. so lets just make one cut from outer edge to circle center ...

now we have the ability to slide one side over the other until a cone is formed...

now that you have done that rotate in your mind the view to the front view this is the view with the center of the "funnel" at the bottom and the base of the single plane triangle in the top of this view. It forms a closed V shape...Are you with me???

now in your mind slide the cut sides in and out until you see that from a base line under the funnel to the outer most edge of the V is 45 deg..subtended...this will make the V -90 deg...

since r was not defined and since it makes no difference anyway lets just say the radius was 4 now at 45 deg the top of the V back to perpindicular with the base is sqrt of 2/2 = .707 this is the percent of the base's radius...so .707 x 4=2.828 ..this gives a total diameter of 5.656 at the cone top a difference of 8 - 5.656= 2.344.....since the circumf. is our key use 3.14159 c for 8=25.13272, c for 5.656 =17.7688 or a difference of 7.364 units..

since we now know how much has been removed from the circle circumf. we can find theta..

25.133 c1/360 deg is to 7.364 d1/X=

105.48 deg...thats theta...

cut from the circle...This angle should give maximum volume for an inverted cone with any radius....(any more and the cone is too steep any less and the cone is to flat) but with a circle radius of 4 units the circumf, cut out is 7.364 units....Long way around the mountain I guess....

now how about some kind of equation..

Well lets use (Optimum Angle) = OA=360-[360*sqrt2/2]

Have a good Thanksgiving....from the E...

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