JUICE-BOX DESIGNS: Contradictions in Minimums
Name ___________________________
High-A is a new juice box company. They guarantee all of their drinks are
100% real juice. Their flagship flavor is the Lobo Lime. Each individual
box will contain 6.75 fluid ounces which is approximately 200 cubic
centimeters. Eight individual boxes will then be packaged in boxes to be
sold at the local supermarkets.
The company wants to reduce the cost in making the drink. The packaging
department has two ways to potentially save money for the company. First,
minimizing the surface area of each individual box should save some
money. Further, minimizing the surface area of the packaged set of eight
will also save the company money.
The target audience for the juice boxes are young children. Thus, the
company wants to make sure little toddlers can easily grip the juice box.
Through research, the company has discovered juice boxes can be gripped
easliy with one hand if the height is exactly twice the width.
Individual boxes:
Your first job is to help the company minimize the surface area of an individual juice box. There are four
requirements that must be met.




Contains 200 cm3 of juice.
The height is exactly twice the width. (h = 2w)
The width measurement in centimeters has to be an integer (1,2,3, etc.).
The length of the box has to be at least 1 cm.
1.
What is the equation for a volume of a rectangular prism?
a) V = lwh
Rewrite this formula in terms of length. l = ? (In other words, get “l” by itself.)
b) How could you rewrite the formula if you know that each box contains 200 cm3 of juice?
2. What is the surface area equation for a rectangular prism?
SA =
3. List out all possible box designs that could be built. Round each length dimension to two decimal places.
Use the information from the bullets in the question. Once the SA hits a low point and then goes starts
going back up, you are done.
w
h = 2w
l
200
wh
SA
1
2
3
4
5
6
7
8
9
10
4. What are the two best possible designs? Identify as Design A and Design B.
Design A:
w = ________
h = ________ l = _________ SA =
Design B:
w = ________
h = ________ l = _________ SA =
Packages of 8 Juice Boxes:
The packaging will be shaped in one of two dimensions: either a 2 by 4 or a 2 by 2 by 2 box. The boxes will
always be standing upright and never on their side. There are three different ways to package the eight juice
boxes as shown below.
Your second task is to find the best way to package eight juice boxes. To do this, you will be trying to minimize
the surface area of the box that will hold the eight individual juice boxes. To find this minimum, you will need
to find the surface area for six possible designs: the three options shown above for Design A and the same three
options for Design B. Find the surface area of the shipping boxes for each of these designs.
Packages of 8 Juice Boxes
Use the pictures on the instructions.
5. Find the minimum Surface Area for Design A and Design B.
Design A:
2 x 2 Double Stacked
2x4
4x2
L_______
L = ________
L = _________
W______
W = _______
W = _______
H = ______
H = _______
H = _______
SA = _____
SA = ______
SA = _______
Design B:
2 x 2 Double Stacked
2x4
4x2
L_______
L = ________
L = _________
W______
W = _______
W = _______
H = ______
H = _______
H = _______
SA = _____
SA = ______
SA = _______
Conclusions:
6. What is the minimum surface area for packaging 8 boxes of Design A? __________________
Design B? _________________
7. Look back at Questions 4 and 6. What is the contradiction?
Cost:
8. Individual juice box: $0.0002 per square centimeter.
How much does one juice box cost to manufacture?
Design A
Design B
9. There will be 8 boxes shipped together at a time.
How much does it cost to manufacture 8 juice boxes?
Design A
Design B
10. Box that contains 8 juice boxes for shipping: $0.0001 per square centimeter.
How much does one shipping box (that will contain 8 juice boxes) cost to manufacture?
Design A
Design B
11. What is the total manufacturing cost of creating 8 individual boxes and their shipping box?
Design A
Design B
12. All things considered, which design and shipping configurations should you choose?