Shape It Up, Ship It Out!
Authors: Judy Peterson and Linda Stephani
Date: June 2008
Written as part of the BITL IV grant, which was funded by the Montana Office of the
Commissioner of Higher Education.
Abstract:
In this activity, the students use volume and surface area calculations and 3-D models to
determine the most efficient and cost effective way to package and ship three speaker
components: a cone, a cylinder and a rectangular prism.
Level: 8th grade
Focal Point: Geometry and Measurement
Time: 2 class periods (90 minutes each)
Materials: centimeter graph paper
Objectives: Students will
1. Create 3-D models of cones, cylinders, and rectangular prisms
2. Calculate the volumes of a cone, cylinder, and rectangular prism
3. Calculate the surface areas of a cone, cylinder, and rectangular prism
4. Select optimal shipping boxes and packing materials to minimize costs and maximize
profits
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Teacher Notes:
In this activity, the students are simulating a money-making effort. They earn $25 per component
they ship to a recycling center. Their goal is to determine the most cost effective way to ship the
three speaker components pictured below. In determining the best selection of shipping box(es)
and the amount of bubble wrap needed to protect the components, they will calculate the volume
and surface area of each component, using previously investigated calculation strategies. Once
they have calculated the cost of packing materials, they will then investigate the shipping costs
for USPS and UPS to determine the most cost effective means of shipping the speakers.
6 in
5 ½ in
4.25 in
13 1/8 in
9 in
13.5 in
16 1/4 in
Weight: 4.5 lb
1.
Weight: 9.25 lb
Weight: 13.25 lb
Skit
To engage the students and set the scenario of the simulation, the students will perform the skit,
“Bummed Out”.
Skit Scenario: (3 teens meeting at their lockers before school –
Steven has just been given an awesome gift from his grandparents
– a red 1970 Ford Mustang --- however, there’s a catch.)
Note: This skit can be done as a Reader’s Theatre or, to get more
students involved, it can be done in groups of three.
2.
Create 3-D Models
The students will use centimeter graph paper to create 3-D models of the speaker components,
where 1 cm is equivalent to 1 in. The most common model for the truncated cone is a cone;
however, the students may explore alternative models.
Hint: For the cone use a circle to create the
cone.
Shape It Up, Ship It Out
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Calculate Volumes
The students will calculate the volumes of the cone, cylinder and rectangular prism (box), using
previously investigated strategies (e.g. building with cubes, filling polyhedron models and using
formulas). For reference, the formulas are given.
Formulas
Bh
3
Cylinders V  Bh
Cones V 
Rectangle Prisms V  Bh or V  lwh
B = area of base
h = height of cone
B = area of base
h = height of cylinder
B = area of base
h = height of rectangular prism
Calculate Surface Areas
The students will calculate the surface areas of the cone, cylinder and rectangular prism (box),
using previously investigated strategies (e.g. creating 2-D nets, and using formulas). For
reference, the formulas are given.
Formulas
Cones SA  B  .5 pl
B = area of base
p = circumference of circle
h = height of cylinder
r = radius
l = slant height, l  h 2  r 2
B = area of base
h = height of cylinder
p = circumference of circle
B = area of base
h = height of rectangular prism
p = circumference of circle
Cylinders SA  2 B  ph
Rectangle Prisms SA  2 B  ph
Speaker Component Volume and Surface Area Answers
Component
Volume (in3)
Surface Area (in2)
Cone
40.0 in3
77.2 in2
Cylinder
858.4 in3
508.7 in2
Rectangular Prism
1173.0 in3
749.7 in2
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Determine Optimal Shipping Box(es) and Create 3-D Model(s)
Using the selections in the Packaging Costs table, the students will select a shipping box or
boxes for their components. Their goal is to minimize their costs and maximize their profits. To
ensure that their components will fit in their box or boxes, the students will build 3-D models of
their choice(s).
Packaging Costs
Boxes
10” x 10” x 14”
$2.49
10” x 12” x 15”
$2.89
14” x 12” x 16”
$2.69
12” x 12” x 12”
$1.89
18” x 12” x 10”
$2.99
18” x 18” x 18”
$3.99
Packing Materials
Bubble Wrap, 3/16” thick,
$11.99
24” x 25’
Peanut Filler, 1.0 cubic feet $4.29
Calculate Packing Material Amounts and Costs
UPS advises that peanut filler is not sufficient to protect multiple items shipped in one box. They
recommend using a combination of peanut filler and bubble wrap. Thus, in this simulation, the
students will use a combination of both. To determine the amounts of bubble wrap, the students
will use their surface area calculations. The amount of peanut filler is determined by subtracting
the total volume of the component(s) from the volume of their shipping box(es).
Note: The price of the peanut filler is given for 1 cubic foot. The students need to convert this
cost to “cost per cubic inch”.
Calculate and Compare Shipping Costs
The students will use the following links to determine the shipping costs for the package(s) using
UPS and USPS. They will compare the costs and determine the most cost effective means of
shipping their components.
Links:
Calculate UPS Shipping Cost
Calculate USPS Shipping Cost
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Calculate Total Cost
The students will calculate the total cost for shipping box(s), packing material, and shipping
costs.
Calculate Total Profit
The students will calculate their total profit, subtracting their total cost from $75.
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