SOLUTION: A cylinder is inscribed in a sphere with radius 6. Find the height h of the cylinder with the maximum possible volume?
Thank you
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Question 999834: A cylinder is inscribed in a sphere with radius 6. Find the height h of the cylinder with the maximum possible volume?
Thank you
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
A sphere radius that intersects the edge of the cylinder base is the hypotenuse of a right triangle with legs that are cylinder radius and one-half of the cylinder height. Given a sphere with radius 6, the radius and height of the cylinder are related thus:
^2)
The volume of the cylinder is:
Substituting:
\ =\ \pi\left(36\ -\ \left(\frac{h}{2}\right)^2\right)h)
Simplifying:
\ =\ \pi\left(36h\ -\ \frac{h^3}{4}\right))
Take the first derivative
\ =\ \pi\left(36\ -\ \frac{3h^2}{4}\right))
Set the first derivative equal to zero and solve for the value of the height that gives the maximum volume.
John
My calculator said it, I believe it, that settles it
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