Expert: Chen Min - 3/15/2011

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
Firstly, you get the arc length corresponding to the sector cut out: R(theta)
It's the circumference of the cone's base: so 2(pi)r=R(theta), r = R(theta)/2(pi), where r is the radius of the cone's base.(You should express r with R and not the other way round)
Note r and R and cone's height h form a right angle, so h=sqrt(R^2-r^2)=R*sqrt[1-(theta)^2/4(pi^2)]
The volume is thus (1/3)*(pi)*r^2*h=(pi/3)*R^3*(theta)^2/(4pi^2)*sqrt[1-(theta)^2/4pi^2]
                                  =(R^3)/(12pi)*(theta)^2*sqrt[1-(theta)^2/4pi^2]

The maximum happens when (theta)^2*sqrt[1-(theta)^2/4pi^2] is biggest;which is equivalent to when (theta)^4*(1-(theta)^2/4pi^2)is the biggest.This is easily computable via mean inequality theorem.
It's at when (theta)^2/8pi^2=1-(theta^2)/4pi^2, i.e. theta=sqrt(8/3)pi
The shape is not really clear..oops...but I don't think my calculation is wrong...Plz check.

The sum of two cones part is rather tricky...I haven't found any easy solutions, as the expression itself is already complex. There must be another way in. You may wish to ask other experts. I'm sorry.