Ashley M.

asked • 05/04/18

Calculus help

A right circular cone is circumscribed about a sphere of radius a. if θ is a semiangle at the vertex, find θ when the volume of the cone is a minimum. Note: The semiangle of a cone is the angle at the vertex between the outer surface of the cone and the line that connects the vertex to the center of the base.

Paul M.

tutor
I am unsure about this problem for the following reason.
 
First, a right circular cone is a solid of revolution of an isosceles triangle and an inscribed sphere is also a solid of revolution.  Therefore, it suffices for this problem to consider the triangle and the circle which when revolved form the cone in question.
 
Once the angle θ is fixed, the triangle is fixed up to similarity, i.e.  the leg lengths may vary only in such a way that such variation produces similar triangles.
 
It is easy to show geometrically that there is only one circle which can be inscribed in the triangle, again up to similarity.  The proof can be shown by showing that the bisector of one of the base angles intersects the perpendicular bisector of the base at the center of the inscribed circle.  
 
At least as far as I can tell there can be no minimum volume.  Knowing either the cone or the sphere forces the other.
 
If some other tutor contradicts this comment, I would be very happy to know the reasoning.  Equally if your instructor can give another answer to this problem, I would very much like to see.
 
Thank you.
 
 
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05/04/18

Doug C.

I have solved problems like this: Find the dimensions of a right circular cone that circumscribes a sphere with radius "a", that also has minimum volume. If that problem is solved then θ = arc tan (r/h). 
 
Solving the original problem is quite complicated. My guess if that if you do a search for:
"A right circular cone is circumscribed about a sphere of radius a" there are some videos out there showing how to find r and h such that the cone has a minimum volume. Remember that "a" is a constant. The problem could just as easily have stated where sphere has radius = 7. My recollection is that finding a relationship between radius of sphere and radius/height of cone involves similar triangles. 
 
As a matter of fact it is coming back to me. My recollection is that the dimensions that minimize the volume (in terms of "a) are r= a√2 and h = 4a. Try using those values for the arc tan suggestion and see what you get for θ.
 
 
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05/05/18

1 Expert Answer

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Paul M. answered • 05/05/18

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B.S. in Mathematics & M.D.
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First of all, I thought I had added a long comment to this answer.  The comment is only partially correct. 
 
The circular cone and the sphere are solids of revolution and for purposes of this problem, only the triangle and circle from which the cone and sphere are constructed by revolution are needed.  Draw a figure of this triangle and circle.
The sides of the isosceles triangle are tangent to the circle.  Label the base of the triangle AC and the vertex opposite AC is B. The center of the circle is E.  The perpendicular bisector of AC is BD and the points of tangency are E (on AB) and G on BC.  ∠ABD is θ.
 
FE = a and BE = a csc θ
BD = a + a csc θ = a(1 + csc θ) = height of the cone
 
AD = BD tan θ = a(1 + csc θ)tan θ = the radius of the cone
 
The volume of the cone = (π/3)r2h = (π/3)a3[(1 + csc θ)tan θ]2(1 + csc θ)
 
The volume will be minimized when the derivative of volume with respect to θ is equal to 0.
Calculating that derivative will be messy at best.  Graph the equation of volume without the constant term.
 
The graph shows 2 minima between 0 and θ, one about 18.9° and one about 160.4°.
 
I have worked over this problem a long time.  I believe this solution is correct, but another tutor or your instructor may have a better solution or an easier one.  If that turns up, I would certainly like you to notify me so that I can learn something!  Thank you.
 
 
 
 
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