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Kapi'olani Community College
Classic Volume of a Box Problem
. . . extended to
On the left the boxes are arranged by the cube
even integers from 8 to 24.
right is the cost of each box from $2.50 to $7.85.
This same set of data may be used as an Excel exercise.
Comparison of the two graphs invites closer inspection of the
data. Note the increment between $3.60 to $3.85 and $5.50
to $5.75 is the same, but also the smallest of the entire data
set. This produces a "kink" in the curve.
The graphing calculator was programmed to produce a smooth curve;
whereas "Excel" simply connects the points. Interesting!
For more ideas and activites see the following:
Coll, Davis, Hall, Magnant, Stankewicz and Wang, "Integer Solutions to
Box Optimization Problems," The
College Mathematics Journal, MAA, Vol. 45, No.3, May, 2014,
quick and interesting graphing calculator
application of exponential
all students in advanced algebra
and calculus have solved a "volume of the box" problem. This
activity uses boxes and a graphing calculator to illustrate
L1 is the list of one dimension of each box.
L2 is the cube of L1, or the volume of each
L3 is the cost of each box.
L4 is the average cost per cubic inch or
After plotting the points ( L1, L4 ) . . . , and noting the
we determine the
exponential regression equation.