Lord of rings ?

 

     A magic ring is made by cutting the sphere in Figure 1 by drilling a cylindrical hole (together with the top and bottom caps). The ring is shown in Figure 2. The only given is that the cylinder is of height 2h. What is the volume of this magic ring?

 

      

 

 

Analysis

 

             Since the radius of the sphere and the radius of the cylinder are not given, it seems that the volume of the magic ring cannot be found.

 

Calculation

Let R be the radius of the sphere. We don’t know R, but we like to carry on.

The sphere is got by rotating the circle:

             C:    x2 + y2 = R2   about the y-axis.

 

Let  x1 be the inner radius (of the cylinder) and x2 be the outer radius (of the sphere) of the elementary disc as shown in the yellow part of Figure 3. The thickness of the disc is dy.

 

Volume of the magic ring

            

Now, x2 is coming from the sphere by rotating the circle C and therefore

             x22 = R2 – y2                     (2)

Also, x1 is coming from the cylinder, which is constant at different height of the cylinder.

             x12 = OQ2 = OP2 – PQ2 = R2 – h2            (3)

 

Substituting (2), (3) in (1), we get:

            

                    

                    

                    

You can then see the final answer is independent of R, the radius of the sphere.

Haha! The volume of the magic ring can be found!

 

 

Quick solution

 

             Since the question gives you only the height (2h) of the ring and not the radius of the cylinder. Therefore we can put the radius of the cylinder tending to 0. Since there is nothing cut away, the magic ring becomes a sphere. This sphere has diameter equal to the height of the cylinder 2h. The radius of the sphere is thus h.

 

             By the sphere formula, the volume of the magic ring

                    

 

 

Final discussion

 

(1)     There are two things that needs consideration for any solution to a problem:
         (a) existence - whether there is/are solution(s).
         (b) uniqueness - whether there is only one solution.

         For the quick solution above :
         (a)       A solution is got by putting the radius of the cylinder to zero.

                   There is no problem for "existence". Fine.
         (b)      If the question states that there is only 'one' solution,

                   the quick solution is perfect.

                   But there are sometimes many solutions to one problem....

 

(2)     If the radius of the sphere is bigger, the cylindrical hole is also bigger.

         However, the volume of the magic ring is unchanged!

         You can draw two diagrams with different radii of the sphere.

         Compare and see what happens.