How to Construct a Cone of Maximum Volume from a Circle
Cones are useful geometric shapes; they make great funnels and containers, and are very easy to construct. If you have a piece of paper in the shape of a circle, all you need to do is cut out a wedge and tape the edges of the remaining circle together. Depending on the size of the wedge you cut out, your cone will have less or more volume. Use a short series of steps to cut out a wedge and create a cone with maximum volume.
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Instructions
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Locate the center of the circle by folding it in half along two different lines, and then noting where the folded lines intersect.
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With the ruler, draw a straight line from the center to the outside edge of the circle. This is one side of the wedge you will cut out to make the cone.
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Using the protractor, draw another line 66 degrees away from the first line. This forms the other edge of the wedge.
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Cut out the wedge, and tape the edges of the circle together to form a cone. By cutting out a sector with an angle of 66 degrees, you have constructed a cone with maximum volume.
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If you are curious, you can verify that 66 degrees yields the maximum volume. If a circle has a radius of r and you cut out an angle of x degrees, the volume of the resulting cone will be:
1.0472r^3(1 - x/360)^2 * sqrt(x/180 - x^2/129600)
Pick a fixed value for r and plug in different values for x between 0 and 180. You will see that x = 66 gives the maximum volume.
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