You are given a circular piece of cardboard, and asked to cut out a sector so that when the cut edges are brought together a cone of maximum volume is formed.

What central angle maximizes the volume of the cone?

a.) Decimal approximation
b.) Exact answer

Question

You are given a circular piece of cardboard, and asked to cut out a sector so that when the cut edges are brought together a cone of maximum volume is formed.

What central angle maximizes the volume of the cone? 

a.) Decimal approximation
b.) Exact answer

datanewb · Answer

I get that the volume V relates to the central angle in this way:$$V=\frac{\pi}{3}sin^2(\alpha)cos(\alpha)$$ ... but taking the derivative has not allowed me to deduce the maximum. Is that equation correct? If so, how does one find the root of $$\frac{\pi}{3}(2sin(\alpha) cos^2(\alpha) - sin^3(\alpha)) $$

noelgreco · Answer

I didn't use trigonometry in my calculation, but in graphing your formula for V I got a maximum at a different value.

noelgreco · Answer

$$\Theta=\frac{ 6\pi -2\pi \sqrt{6} }{ 3 }$$, or 
1.15299 radians
$$66.06^{o}$$

datanewb · Answer

|dw:1346808180730:dw| oops I misunderstood what a central angle was! Let me recompute!