What is the optimal angle for a conical cup with a maximum volume?

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SUMMARY

The optimal angle for a conical cup with maximum volume, formed from a sector of a circle with a radius of 8 cm, can be determined using the volume formula V = (πr²h)/3. The challenge lies in expressing the height h in terms of the angle θ. By applying similar triangles, the relationship h = 1/(1 - R/r) can be derived, where R is the radius of the cone's base and r is the radius of the sector. Further analysis is required to incorporate the angle θ into the volume equation for maximization.

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A conical cup is to be made by joining the edges OA, OB of the sector of a circle of radius 8cm. What angle @ gives the cup of maximum volume?


I'm having trouble solving this question. This is what I have done so far

we know V=pi*r^2*h/3

r is given, so we need to put h in terms of @.
now by similar triangles we can take another slice, and compare
r/h=R/(h-1)... (h-1) is just the height of the triangle below the new slice
solving for h I get h=1/(1-R/r) now I'm unsure of what to do now, and how to get @ into the equation.


Anyways please let me know if I'm doing this question completely wrong or what not. Thanks for all your help
 
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r is not given! You're given the radius of the sector of the circle but that is not the same as the radius of the cone.
 
a picture would help
 

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