Minimal Volume of a Cone Circumscribed about a Sphere
Find the minimal volume and dimensions of a right circular cone circumscribed about a sphere of a given volume. To solve this problem we need to know
1) The formula for the volume of a sphere
2) The formula for the volume of a cone
3) The radius of a sphere is perpendicular to a tangent line to the sphere.
4) Setting up ratios using similar triangles.
Find the volume, radius, and height of a right circular cone of smallest volume about a sphere of volume .
Let represent the volume of the sphere, x represent the radius of the sphere and , r and h represent the volume, radius and height of the right circular cone. I will solve the problem in general for any volume of a sphere.
1) Determine the radius of the sphere.
2) Set up similar triangles using the geometry of the problem and represent the volume of a cone a function of h.
Squaring each side gives
Now solve for in terms of h.
, cross multiplying
, move all terms containing to the right
, factor out
, foil, combine like terms then divide each side by
Now substitute this value for into formula for the volume of the cone
I have parted to make taking the derivative easier.
3) Take the derivative of with respect to h using the quotient rule.
4) Set and solve for h.
Note that the derivative does not exist at h=2x but 2x and 0 is not in the domain so the only possible value for h is 4x.
Now solve for the radius.
5) Solve for the volume of the cone representing all values in terms of .
Substituting in the numerical value of we have
About The Author
|College Math, Statistics And Trigonometry|
|I am a graduate of Cleveland State University. After earning my Bachelors of Electrical Engineering I went on to earn a Master’s of Science in Electrical Engineering. Also I am a graduate of Lorain County Community College (LCCC) earning an Associates of Science Pre-Professional Engineering and ...|