A BEAM OF MAXIMUM
VOLUME
The
problem is to saw out the largest rectangular beam from a cylindrical log. Find
the shape of the cross section it will have
 |
If
the sides of the rectangular cross section are x and y then by Pythagoreantheorem,
we have
Where
d is the diameter of the log. The
volume of the beam is a maximum when the area of its cross section is a
maximum. That is, when xy becomes a maximum. Now if xy is a maximum, then so is
the product 
. Since the sum 
is constant, it follows by what has been
already proved that the product 
is the largest possible one when
. Since the sum is constant, it follows by what has been
already proved that the product is the largest possible one when 
or
Hence
the cross section of the beam must be a square
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