Maximizing the Volume and Surface Area of Geometric Solids Inscribed in a Sphere

figure

cylinder

cone

cuboid

objective

volume

surface area

r

show projection

r = 0.5000

volume = 1.3603

surface area = 7.0122

This Demonstration illustrates two common types of max-min problem from a Calculus I course—those of finding the maximum volume and finding the maximum surface area of a geometric figure inscribed in a sphere. The figures available are a cylinder, a cone, and a cuboid with a square base. The sphere has radius 1. For the cylinder and the cone, r is the radius of the base. For the cuboid, r is one-half the length of a side of the base. An option to show the projection of the sphere and the inscribed solid onto the

x

-

z

plane is included (see snapshot 5).