CPhill

Thanks, killersteve   !!!

I'm making one assumption in this problem...I'm assuming that when the larger cylinder is on the bottom [the natural position ]....the molasses fills all of that cylinder  and none of the top cylinder...this would seem to be logical

Let us call the height of the smaller cylinder, h

First....when inverted so that the smaller cylinder is on the bottom, the molasses completely fills the smaller cylinder and the height of molasses that fills the larger cylinder  is just  (19 + 1/3 - h)  =  (58/3 - h)

So.....the total volume, V, of the molasses in this position  is just

pi (2^2)h  + pi (6^2) (58/3 - h)  = V

4h * pi  +  36pi (58/3 - h)  =  V

[4h + 36 ( 58/3 - h)] pi  =  V

[ 4h + 696 - 36h ] pi  = V

[696 - 32h] pi    =  V   (1)

And when the can is in its natural position, it  fills all of the larger cylinder...so....the volume can be expressed as :

14* 36 * pi =  V

504 pi  =  V    (2)

Equate (1)  and (2)

504  =  696 - 32h

32h   = 192

h  = 6 cm  = the height of the smaller cylinder

So.... the total height of the can [ under my assumption]   =  6cm + 14cm  =  20 cm

This is analogous to filling  N distinct boxes with K identical balls

If  there are no restrictions as to the number of plants each sill can contain ....the number of possible  ways   is

C( K + N - 1, N - 1)  = 

C ( 6 + 3 - 1, 3 - 1)  =

C (8, 2) = 28 ways

If each sill must contain at least one plant, the number of possible ways is

C( K - 1, N - 1)  =

C( 6 - 1, 3 - 1)  =

C(5, 2)  =  10 ways

Impressive, hectictar  !!!!

Scroll down  all the way to the bottom of the forum page and look at the right side....

Thanks....we always appreciate those compliments....!!!

m=2(nx+bn+bx)

I think you wanted to solve for x

Divide both sides by 2

(m /2)  =  nx + bn + bx     subtract  bn from both sides

(m / 2) - bn  =   nx + bx     simplify the left side as a single fraction

[ m - 2bn ] / 2     =  nx + bx      factor the right

[ m - 2bn ] / 2   =   x [ n + b]        divide both sides by  [ n + b ]

[ m - 2bn]  / [ 2 ( n + b) ]  =  x

I don't know that this can be expressed as a single function....but....as a piecewise "function"  we might try this

y  =  x          1 ≤ x ≤ 4     where x  is an integer

y  =  -x + 8     5 ≤ x ≤ 8    where x is an integer

If it's in one variable  (usually x)....look for the highest power on any x....this will be the degree....

tan (-pi)  =  tan (pi)   =  0