THE PUZZLET PAGE

PUZZLET 018 - THE SOLUTION

This week's Puzzlet is an interesting study in controlling dimensions. The solution depends on developing a formula that captures the required variables in a useful way.  The code below describes the problem in the introductory remarks, and will produce the solution when run.

' Puzzlet #018
' A lid-less tank is a cube in shape.
' All external dimensions are 1 metre.
' It is designed to be filled to within
' 10 cm of the tank's top, when the
' ratio of the tank material to its
' contents will be exactly 5%.  What is
' the thickness of the tank's material
' in mm to 2 decimal places of accuracy?
' By Dave Ellis.

def flag: int
def func, diffFunc: float
def lastRoot, root: float
root = .02
flag = 0
openconsole

do
    func = 84*root*root*root - 167.6*root*root + 104.6*root- .9
    diffFunc = 252*root*root - 335.2*root + 104.6
    lastRoot = root
    root = root - func/diffFunc
    if abs(lastRoot - root) < .0001
        print "Material Thickness: ",
        print using ("##.##", 1000*lastRoot),
        print " millimetres."
        flag = -1
    endif
until flag = -1

print: print "That's all, folks!"
print: print "Press any key to exit ..."
do: until inkey$ <> ""
closeconsole
end

PROGRAM NOTES

Puzzlet 18 set us to work designing a Mixing Tank.  It needs a base, 4 sides, but no top.   External dimensions are all 1 metre exactly.   It will be filled with fluid to a line 10 cm from the top.  The volume of material used to make the tank must not exceed 5% of the volume of fluid in the Tank.  What is the thickness of the material used, to the nearest tenth of a millimetre?

This poser needs a little analysis to derive its equation. Since it's in Puzzlet, you've no doubt guessed we need a computer program to solve the equation.

So, what do we know?   Firstly, the volume of the tank overall  is 1 cubic metre.  Secondly, we can express its internal volume and permitted fluid volume in terms of material thickness.  Finally, we can use the above to derive a formula for material volume.  Since the material volume is 5% (maximum) of the fluid volume, we can combine these two and develop the final equation ready for the computer to do the donkey work.  This is how it all goes together, using the abbreviation t for thickness:

Internal base width  = (1 - 2*t) metres.
Internal base depth  = (1 - 2*t) metres.
Internal max height  = (1 - t) metres.

Therefore internal volume =  (1 - 2*t)2(1 - t)
                                               
= (1 - t)*(1 - 4*t + 4*t2)
= 1 - 5*t   + 8*t2 -  4*t3 cubic metres.

Now we can find material volume, since it is  total  volume minus internal volume.   We've already seen that total volume is 1 cubic metre.  Therefore, material volume is:

1 - ( 1 - 5*t   + 8*t2 -  4*t3 )
=  4*t3 - 8*t2 + 5*t cubic metres.

On to fluid volume,  bearing in mind that the fluid height is 10 cm below the top of the Tank.

Internal  base width = (1 - 2*t) metres.
Internal base depth = (1 - 2*t)  metres.
Fluid height  = (1 - t -  0.1) = (0.9 - t) metres.

Therefore fluid volume =   (0.9 - t)*(1 - 2*t)2

=  0.9 - 4.6*t + 7.6*t2 - 4*t3 cubic metres.

So now we have equations for fluid volume and material volume.   We know that the latter must not exceed 5% of the  former.   This  can be expressed as:

Fluid volume (minimum) = 20 * material volume.

Substituting the formulae  derived above into this equation:

0.9 - 4.6*t + 7.6*t2 - 4*t3 = 20*( 4*t3 - 8*t2 + 5*t)

Rearranging and multiplying out gives:

84*t3 - 167.6*t2 + 104.6*t - 0.9 = 0

And so we come back to one of our old favourites, the cubic equation. We've encountered this before several times in Puzzlets, and  used various ways to solve it.   On  one  occasion,  Newton's Rule was used, and that's the one deployed here.   If you want an explanation of how this neat mathematical toy works, it's in the Answer to Puzzlet 5, which you can find by going to the Archives Page.
 

When you run the code,  it will produce the inset screen.  So, the answer to the nearest tenth of a millimetre is 8.7
 Material Thickness:  8.73 millimetres.

That's all, folks!

Press any key to exit ...
          

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Last Updated: February 22nd, 2004.