# Mov Largest volume of a cylinder inscribed in a sphere

Sumit Nov 20, 2013 By:

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**Last modified:** 03 May 2013

"Find the largest possible volume of a right circular cylinder that can be inscribed in a sphere with radius r."

Learn how to find the largest possible volume of a cylinder inscribed in a sphere with radius r. To solve this optimization problem, draw a picture of the problem and label all parts of the diagram, then write down everything you know. Next, identify optimization and constraint equations. The optimization equation will be the equation for the volume of the cylinder, since the goal is to maximize the volume of the cylinder. The constraint equation will include the variable that constrains you. In this case, the constraint equation will be the equation for the radius of the sphere. Solve the constraint equation for one of the variables, and then plug the result into the optimization equation. Then simplify the optimization equation, take its derivative, and set it equal to zero. Solve for the variable, and then plug that back into the equation for the volume of the cylinder to find the largest possible volume of the cylinder that can be inscribed in a sphere with radius r.

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