What is the height of a circular cone with surface

Sattelhofsk

Sattelhofsk

Answered question

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?

Answer & Explanation

Jaida Sanders

Jaida Sanders

Beginner2022-06-21Added 18 answers

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 [ 0 , s ], and the radius of the bottom circle will then be r = s 2 h 2 . Therefore we have to maximize the function
V ( h ) = π 3 r 2 h = π 3 ( s 2 h h 3 ) ( 0 h s )   .
As V ( 0 ) = V ( s ) = 0 this maximum is taken for some h   ] 0 , s [   , which will come to the fore by solving V ( h ) = π 3 ( s 2 3 h 2 ) = 0 for h. There is only one solution in the interval   ] 0 , s [   , namely
h = s 3   ,
and this is the height you wanted to know.

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