The optimum design for a boxlike doghouse.
==========================================

(I screwed this up a little bit in class, so let me do it again here.)

It costs 1 dollar per square meter to buy plywood sheet.
Your mission is to build a box containing 1 cubic meter of air,
that costs the least.  The box is allowed to have no floor, since
you get to use a (free) dirt floor.  (There is also no door, since
we want the dog not to get out. Hey, this is calculus class, not
be nice to pets day.)

Let the box be A north-south, B east-west, and H high.

VOLUME(in cubic meters) = 1 = A B H.

SURFACE AREA(in square meters) = COST(in dollars) = 2 (A+B) H + A B
                                            
                                    = AREA(walls) + AREA(roof)

LEMMA: The best house has a SQUARE floor plan, A=B.

PROOF.
If H is fixed, then that means AB is fixed at some value k
(to make the volume be fixed) so B = k/A.  Then
    SurfaceArea(A) = 2(A+k/A)H + k
and this is minimized when
                               -2
    SurfaceArea'(A) = 2(1 - k A  )H = 0
which happens when
         2
    k = A
which happens when A = B   (since B=k/A).
END OF PROOF OF LEMMA.

So great, we can restrict attention to square-floor houses. In that case
the problem is simpler:

            2                               2
  VOLUME = A  H     SURFACE AREA = 4 A H + A 
                                                       -2
and if we have fixed volume(1 cubic meter)  then  H = A  .

That means:
                                      2
        SURFACE AREA = COST = 4/A  + A 

To find the best (least cost) shape we need to solve
    COST'(A) = 0
for A, i.e. solve
         -2                                2
     -4 A  + 2 A = 0    i.e. (multiply by A /2 )
i.e. solve       
           3
   - 2 +  A  = 0

                  1/3   
solution A = B = 2     = 1.26   !!!

          -2    -2/3
Then H = A   = 2      = 0.63.

So that is the best, cheapest, doghouse shape to build!!  (At least,
for a dog that occupies 1 cubic meter, doesn't care what shape it is,
just cares it is 1 cubic meter!)   Actually, a lot of motel 6's seem 
to have those approximate dimensions (?) - coincidence?