Maximizing Volume
March 23, 2004
Profit and Product Distribution . . .
Why does cereal come in a rectangular prism
and potato chips come in a bag?
Manufacturing and Advertising . . .

Why does ice cream come in so many different
containers?

There are so many things that go into packaging
development, one of which is trying to use the least
material (which is cheaper) that will hold the most of
their product while maximizing the surface area to
advertise the product. Thus they are increasing profit
while decreasing cost.
Maximizing Volume
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Suppose cardboard for your product comes in
8.5 x 11 sheets. You want to construct a box
(ignore the top) that will hold the most of your
product.
To construct a rectangular prism from the
sheet, you cut out congruent squares from the
corners, then fold up the sides.
–
What size corners should you cut out to get the
biggest box possible, the one with the most volume?
x=1
x=1
Let x = 1 in.
Cut 1 by 1 in
squares from
each corner.
Fold up the
sides to form
the open box.
What are the
dimensions of
the box?
What is its
volume?
Volume of the box (prism)?
V = AH
What are the dimensions of
the base and its Area?
Length of the base =
(8.5 – 1 – 1) = 6.5
Width of the base =
(11 – 1 - 1) = 9
What is the Height of the box?
1
What is the volume?
V = (6.5)(9)(1) = 58.5
Is this the most volume
we can get?
Group Activity: Use your paper to cut and
then fill in this table.
H
W of base L of base Volume
1 in
9 in
6.5 in
58.5 in3
1.5 in
2 in
3 in
4 in
x
11 – 2x
8.5 – 2x
y=
Homework Quiz
1.
2.
3.
Get out your chart
Get out your box
Get out your graphing calculator
How do we find the “best” box?
http://www.mste.uiuc.edu/users/carvell/3dbox/
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Add a surface area column to your chart!
We use algebra and our calculator to find THE
x that will give us the dimensions for the MOST
volume.
Go to Y= on your calculator and enter our
volume equation: y = x(8.5 – 2x)(11 – 2x)
For our data, we must set our Window to see
the part of the graph we need.
Window and Steps

xmin:0
–
–


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xmax:15
xscl: 1
ymin: 0
–


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square dimension & height of
prism
can’t have negatives



–
volume
ymax: 100
yscl: 10
xres: 1
Then graph
Trace to find max
volume (y) on graph
Use Calc (2nd trace) to
find exact max.
–

left bound is (0,0)
right bound?
What x do we get?
Finally . . .


The maximum volume for a rectangular prism
made from an 8.5 x 11 sheet of paper will be
66.148 in3 and will occur when the squares cut
from each corner measure 1.5854 in x 1.5854
in.
How much is that? Let’s eat!
Classwork and Homework

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Page 532 # 1, 2, 3, 5, 10 (collecting tomorrow)
Study for your test tomorrow.
–
–
–
–
naming prisms, pyramids, cylinders, cones and their
parts (10.1)
finding volumes of those objects (10.2, 10.3)
doing an application problem (10.4)
about 15 problems