Cylinder and cones (Putnam Exam 1938) Right circular cones

Chapter 7, Problem 40E

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QUESTION:

Right circular cones of height h and radius r are attached to each end of a right circular cylinder of height h and radius r, forming a double-pointed object. For a given surface area A, what are the dimensions r and h that maximize the volume of the object?

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QUESTION:

Right circular cones of height h and radius r are attached to each end of a right circular cylinder of height h and radius r, forming a double-pointed object. For a given surface area A, what are the dimensions r and h that maximize the volume of the object?

ANSWER:

Solution 40E Step 1: Consider that the right circular cones of height hand radius rare attached to each end of a right circular cylinder of height h and radius r, forming a double pointed object. Consider the following figure,

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