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Problem
2.2 Follow-Up |
| 1. |
Test
your conjecture from Problem 2.2 on some other examples, such
as 30 cubes or 64 cubes. Does your conjecture work for the
examples you tried? If not, change your conjecture so it works
for any number of cubes. When you have a conjecture that you
think is correct, give reasons why you think your conjecture
is valid. |
| 2. |
What
rectangular arrangement of cubes uses the most packaging material?
Why do you think this is so? |
| 3. |
What
is the surface are of the box below? Explain how you found
your answer. |
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| 4. |
Suppose
the box in question 3 were resting on a different face. How
would this affect its surface area? |
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| Did
you Know?
Area
is expressed in square units, such as square inches
or square centimeters. You can abbreviate square units by
writing teh abbreviation for the unit followed by a raised,
or superscripted, 2. For example, an abbreviation
for square inches is in2, and an abbreviation
for suqare centimeters is cm2.
Volume
is expressed in cubic units, such as cubic inches
or cubic centimeters. You can abbreviate cubic units by
writing the abbreviation for the unit followed by a superscripted
3. For example, an abbreviation for cubic inches is in3,
and an abbreviation for cubic centimeters is cm3.
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Answers
to Problem 2.2 Follow-Up
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| 1. |
See page 23f. |
| 2. |
In general, the arrangement with the greatest surface area
is the one that is most spread out, so it would require the
most packaging material. (Note: This is the 1 by 1 by n
arrangement; surface area is maximized because at least four
faces of every cube are exposed.) |
| 3. |
62 in2; Possible explanation: The surface area
is the sum of the area of the faces. In a rectangular prism,
there are three pairs of congruent faces, so you find the
area of the three faces and double the sum. In this box, two
faces are 5 in by 3 in (15 in2), two are 3
in by 2 in (6 in2), and two are 5 in by 2
in (10 in2; 2(15 + 6 + 10) = 62 in2. |
| 4. |
The surface area of a box is the sum of the area of its
six faces and is the same no matter which face is used as
the base. |
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CONNECTED
MATHEMATICS PROJECT - Designing Packages - Investigation 2
© Dale Seymour Publications® - Lappan, Fey, Fitzgerald, Friel,
and Phillips
MMM Project Web Version © Bolster Education
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