Inscribe Cylinder in a Sphere

While repairing an old washing machine, Wisdom finds a cylinder balance weight needs to be replaced. In order to save money, Wisdom decides to make this part by cutting a steel sphere. He wants the cylinder balance weight heaviest.

What is known:

• The radius of the sphere is 0.05 m.
• The mass density of this sphere is 7,900 kg/m³.
• The cylinder is inscribed in the sphere.
• The weight of the cylinder, w, is determined by its mass density, ρ, and volume, v.

What can the maximum weight the cylinder be?

• Find the volume that can be inscribed in the sphere.
• Using the second derivative test to verify the volume is maximum.