Sattelhofsk

2022-06-20

What is the height of a circular cone with surface line ("Mantellinie") s, which has the maximal volumina? My problem here is, that I am not really sure what surface line is..is this the same thing as just to say surface?

Jaida Sanders

The "Mantellinie" $s$ is what you think it is: It's the distance from the tip of the cone to the rim at the bottom. Given $s$, you are free to choose the height h in the interval $\left[0,s\right]$, and the radius of the bottom circle will then be $r=\sqrt{{s}^{2}-{h}^{2}}$. Therefore we have to maximize the function

As $V\left(0\right)=V\left(s\right)=0$ this maximum is taken for some , which will come to the fore by solving ${V}^{\prime }\left(h\right)=\frac{\pi }{3}\left({s}^{2}-3{h}^{2}\right)=0$ for $h$. There is only one solution in the interval , namely

and this is the height you wanted to know.

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