[Enrich20161103NRP] The Napkin Ring Problem

Problem

     Two rings are made by drilling a cylindrical hole through a small sphere and a hole through the large sphere, such that the resulting rings have the same height (2h).

     Which ring has the larger volume of remaining material?

Solution

     The answer is: both rings have the same volume.  How can we know?

     There is a way to show this using integration.  But calculus is not necessary.

     Let  r  be the radius of any chosen sphere and let  a  be the radius of the cylindrical hole.  By Pythagoras’ Theorem,  h² = r² – a².  Consider a cross-section of the ring sliced a distance  x  from the centre of the sphere, perpendicular to the axis of the cylindrical hole.  The outer radius of this cross section is the square root of  r² – x².  Hence the area of the material in the cross-section is

               p [(r² – x²) – a²]  =  p (r² – a² – x²)  =  p (h² – x²)

Note that  r  does not appear in the formula.  That means the cross-section does not depend on  r.  A bigger (or smaller) sphere would have the same cross-sectional area for each distance  x  away from the centre.  By Cavalieri'sPrinciple, the other sphere will have the same volume!

     By the way what is this volume?  It is the same as that of a sphere without hole  i.e.  where  a = 0  and  r = h.  This works out to be  ⁴/₃ p h³,  where  h  is half the height of the ring.

Heuristics Used

H02. Use a diagram / model

H09. Restate the problem in another way

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘A’ Levels H2 Mathematics (Number patterns, with algebra)

* revision for IB Mathematics HL & SL

* Advanced Placement (AP) Calculus AB & BC

* University / College Calculus

* other syllabuses that involve volumes and Pythagoras’ Theorem

* any learner who is interested

[S2 Expository] Square-of-Difference Identity for Algebra

     An algebraic identity is an equation that is true for all values of the variables involved.  If we substituted any set of values to the Left Hand Side (LHS) and the same values to the Right Hand Side (RHS), the equation will be true i.e. the LHS will always be equal to the RHS.  The square-of-difference identity

                                       

is one of the three identities that students have to learn in secondary two.  Many students have difficulty remembering this, and they mix this up with the other identity, which involves  a² – b².  However  (a – b)²  is not the same as  a² – b².  They do not understand why the above formula is true, because almost nobody explains it.  Perhaps a few teachers explain the identity for  (a + b)².  But if the  ‘+’  is changed to a  ‘–’  this is a little trickier.  Let me try to explain the formula visually, and with colours to boot, for perhaps the first time in history.

     We start (on the left) with a square of side  a,  whose area is  a².  This is shown in green in the diagram.  We partition each side of the square into  a – b  and  b.  Our goal is to get an area of   (a – b)².  Let us flip the strip of width  b  on the right of the square.  This strip has area  ab  and is shown in pink in the middle square.  This is the same as saying we are subtracting one copy of  ab.  Note on the bottom of the square, there is another strip of area  ab  (shown outlined in orange).  If we subtracted that, we would have subtracted  2ab  (see the square on the right), and we would seem to get  (a – b)².   But then the little square of area  b²  (indicated by a darker green) would have been subtracted twice.  So we need to add  b²  back, so as to restore balance in the universe. 

     You can imagine doing this with a square of area  a²  made of layer of sand.  We remove strips of area  ab  two times – from the right and from the bottom.  Then we patch up the  b²  hole by adding back a layer of sand.  We finally end up with a layer of sand of area  (a – b)².  This illustrates why  a² – 2ab + b² = (a – b)².

     Isn’t this kewl?

Suitable Levels

* Lower Secondary Mathematics

* other syllabuses that involve algebra, expansion and factorisation