Question 1181729
.
The diameter of a sphere coincides with the axis of a right circular cone, and the surface 
of the cone intersects the surface of the sphere in in a great circle. Find: 
(a) the vertical angle of the cone: 
(b) the volume of the cone; 
(c) the volume common to the two solids: 
(d) the volume of that portion of the cone which lies outside the sphere: 
(e) the volume of that portion of the sphere which lies outside the cone. Denote radius of sphere by R.

Solve Problem if (a) R=2: (b) R=3.72.
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The solution in the post by @CPhill is INCORRECT.


It is incorrect, since @CPhill incorrectly interprets the problem.


He solves the problem in assumption that the cone TOUCHES the sphere 
along a great circle.


But, firstly, it is {{{highlight(highlight(IMPOSSIBLE))}}}, and, secondly, the problem {{{highlight(highlight(tells))}}} something {{{highlight(highlight(DIFFERENT))}}}.


It says that the cone' surface {{{highlight(highlight(INTERSECTS))}}} the sphere surface at the great circle,
which is totally different condition.


Thus, the value of @CPhill' solution is ZERO.


He is just deceiving you.



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                Regarding the post by @CPhill . . . 



Keep in mind that @KPhill is a pseudonym for the Google artificial intelligence.


The artificial intelligence is like a baby now. It is in the experimental stage 
of development and can make mistakes and produce nonsense without any embarrassment.



                It has no feeling of shame - it is shameless.



This time, again,  it made an error.



Although the @CPhill' solution are copy-paste  Google  AI solutions,  there is one essential difference.


Every time,  Google  AI  makes a note at the end of its solutions that  Google  AI  is experimental
and can make errors/mistakes.


All @CPhill' solutions are copy-paste of  Google  AI  solutions, with one difference:
@PChill never makes this notice and never says that his solutions are copy-past that of Google.


So,  in my view,  doing this way,  this guy,  @CPhill,  simply makes dishonest business at this forum.



Every time,  @CPhill embarrasses to tell the truth.

But I am not embarrassing to tell the truth,  as it is my duty at this forum.



And the last my comment.


When you obtain such posts from @CPhill,  remember,  that  NOBODY  is responsible for their correctness, 
until the specialists and experts will check and confirm their correctness.


Without it,  their reliability is  ZERO and their creadability is  ZERO,  too.



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Special note for a visitor who posted this problem.


It borders on madness to pack 5 (five) questions into one post.


The rules of this forum allow one question per post.


Even if assume that the rules are made of rubber, then 3 questions per post is the maximum.


Every time, as you violate this rule, you EITHER will wait several years, OR will get a mess.


Regarding the rules of this forum, see this page


https://www.algebra.com/tutors/students/ask.mpl?action=ask_question&topic=Equations&return_url=http://www.algebra.com/algebra/homework/equations/


from which you post your problems.



It is assumed that you read these rules before posting.


It is also assumed that you do understand what is written in that page and follow the rules.



        Those who violate them,  work against their own interests.