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Advanced Math/Precalculus - Volume of Cone Related Rates

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Expert: - 3/13/2011


Question
A cone is made by cutting a sector of angle (theta) from a circle with a radius of 1 and gluing the edges together.

What are expressions for the volume of the cone?
Which angle of (theta) gives maximum volume?
What shape is the maximal cone?

Lastly,

The sector not used is part 1 is also used to make a cone
Which angle of (theta) maximises the sum of the volumes of the cones

Thanks for the help, im a bit stuck, I tried solving it using related rates of change but I don't think that was the right way to approach it.

Answer
The volume of a cone is πr²h/3 where r is the raidus and h is the height.

To find the maximum volume, do you mean with respect to the surface area?
Does this include the bottom?

The sides of the cone have area πrs, where s is the length of the slope on the outside.
This is found by s² = √(r²+h²), where h is the height.  The bottom of a cone is a circle,
and the area of that is πr².  This makes the outside area πr(r+√(r²+h²)).

To find the maximum volume with respect to the surface area, make the surface area a constant,
then solve this equation for surface area for h in terms or r, put that in the volume equation,
take the derivative, and set it equal to 0.

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