Calculus I Project - Optimization

Group project using calculus, determine the optimized surface area and volume of day-to-day cans. Graph to compare the volume vs radius and surface area vs radius. Draw your analysis based on the calculations.

1. Daniel Oliver, Matt Werno, Casey Byrnes, Rebecca Fenter

Tuna Can

Measured Dimensions:
Radius: 4.25 cm
Height: 3.4 cm
Content Volume: 118.29 �??????
3

Actual Volume: ??????(4.25 �??????)
2
(3.4 �??????)=
61.41?????? �??????
3
,�?????? 192.93 �??????
3


??????=????????????
2
ℎ
??????=2????????????ℎ+2????????????
2

Optimizing Volume
??????=2??????(4.25)(3.4)+2??????(4.25)
2
=65.03?????? �??????
3

65.03??????−2????????????
2
2????????????
=
65.03−2??????
2
2??????
=ℎ

??????=????????????
2
(
65.03−2??????
2
2??????
)=
65.03????????????
2
−????????????
3
=32.51????????????−????????????
3

�??????
�??????
=32.51??????−3????????????
2
=0
32.51??????
3??????
=??????
2

→ ??????=±3.29 �??????
65.03−2(3.29)
2
2(3.29)
=6.58 �??????=ℎ
??????
���????????????�??????=??????(3.29)
2
(6.58)
=224.16 �??????
3
,�?????? 71.35?????? �??????
3

%=
61.41??????
71.35??????
=0.8607,�?????? 86.07% ��??????????????????????????????��

Optimizing Surface Area
61.41??????=????????????
2
ℎ
61.41
??????
2
=ℎ
??????=2????????????(
61.41
??????
2
)+2????????????
2
=
122.83??????
??????
+2????????????
2

�??????
�??????
=
−122.83??????
??????
2
+4????????????=0
4????????????=
122.83??????
??????
2

→ ??????
3
=
122.83??????
4??????



→ ??????=3.13 �??????
ℎ=
61.41
??????
2
=
61.41
(3.13)
2
=6.26 �??????
??????
���????????????�??????=2??????(3.13)(6.26)+2??????(3.13)
2
=184.84 �??????
2

2. Daniel Oliver, Matt Werno, Casey Byrnes, Rebecca Fenter

Tuna Can: Graph: Volume vs Radius

Tuna Can: Graph: Surface Area vs Radius

3. Daniel Oliver, Matt Werno, Casey Byrnes, Rebecca Fenter

Tuna Can Analysis
Part 1:
The tuna can had a radius of 4.25 cm, a height of 3.4 cm, and a volume of 192.93 cm
3
. The
volume was 87.1% optimized. The can could have been designed to hold more volume by
decreasing the radius to 3.29 cm and increasing the height of the can to 6.58 cm. These
dimensions would have the same surface area, but able to hold 224.16 cm
3
.
Part 2:
The manufacturer did not make this can the cheapest way possible. They could have saved
money by increasing the height to 6.58 cm and decreasing the radius to 3.29 cm. A can of these
dimensions would have a reduced surface area yet still hold the same volume as the current
design. This would cost the manufacturer less money for each can.

4. Daniel Oliver, Matt Werno, Casey Byrnes, Rebecca Fenter

Chicken Can

Measured Dimensions:
Radius: 5 cm
Height: 5.5 cm
Content Volume: 354.88 �??????
3

Actual Volume: ??????(5 �??????)
2
(5.5 �??????)=
137.4?????? �??????
3
,�?????? 431.97 �??????
3


??????=????????????
2
ℎ
??????=2????????????ℎ+2????????????
2

Optimizing Volume
??????=2??????(5)(5.5)+2??????(5)
2
=105?????? �??????
3

105??????−2????????????
2
2????????????
=
105−2??????
2
2??????
=ℎ

??????=????????????
2
(
105−2??????
2
2??????
)=
105????????????
2
−????????????
3
=52.5????????????−????????????
3

�??????
�??????
=52.5??????−3????????????
2
=0
52.5??????
3??????
=??????
2

→ ??????=±4.18 �??????
52.5−2(4.18)
2
2(4.18)
=8.37 �??????=ℎ
??????
���????????????�??????=??????(4.18)
2
(8.37)
=459.98 �??????
3
,�?????? 146.42?????? �??????
3

%=
137.4??????
146.42??????
=0.9391,�?????? 93.91% ��??????????????????????????????��
Optimizing Surface Area
137.5??????=????????????
2
ℎ
137.5
??????
2
=ℎ
??????=2????????????(
137.5
??????
2
)+2????????????
2
=
275??????
??????
+2????????????
2

�??????
�??????
=
−275??????
??????
2
+4????????????=0
4????????????=
275??????
??????
2

→ ??????
3
=
275??????
4??????



→ ??????=4.10 �??????
ℎ=
137.5
??????
2
=
137.5
(4.10)
2
=8.19 �??????
??????
���????????????�??????=2??????(4.10)(8.19)+2??????(4.10)
2
=316.34 �??????
2

5. Daniel Oliver, Matt Werno, Casey Byrnes, Rebecca Fenter

Chicken Can: Graph: Volume vs Radius

Chicken Can: Graph: Surface Area vs Radius

6. Daniel Oliver, Matt Werno, Casey Byrnes, Rebecca Fenter

Chicken Can Analysis
Part 1:
The chicken can had a radius of 5 cm, a height of 5.5 cm, and a volume of 431.97 cm
3
. The
volume was 93.9% optimized. The can could have been designed to hold more volume by
decreasing the radius to 4.18 cm and increasing the height of the can to 8.37 cm. These
dimensions would have the same surface area, but able to hold 459.98 cm
3
.
Part 2:
The manufacturer did not make this can the cheapest way possible. They could have saved
money by increasing the height to 8.19 cm and decreasing the radius to 4.10 cm. A can of these
dimensions would have a reduced surface area yet still hold the same volume as the current
design. This would cost the manufacturer less money for each can.

7. Daniel Oliver, Matt Werno, Casey Byrnes, Rebecca Fenter

Red Bull Can

Measured Dimensions:
Radius: 2.5 cm
Height: 15.8 cm
Content Volume: needed �??????
3

Actual Volume: ??????(2.5 �??????)
2
(15.8 �??????)=
98.75?????? �??????
3
,�?????? 310.23 �??????
3


??????=????????????
2
ℎ
??????=2????????????ℎ+2????????????
2

Optimizing Volume
??????=2??????(2.5)(15.8)+2??????(2.5)
2
=91.5?????? �??????
2

91.5??????−2????????????
2????????????
=
91.5−2??????
2
2??????
=ℎ

??????=????????????
2
(
91.5−2??????
2
2??????
)=
91.5????????????
2
−????????????
3
=45.75????????????−????????????
3

�??????
�??????
=45.75??????−3????????????
2
=0
45.75??????
3??????
=??????
2

→ ??????=±3.91 �??????
45.75−2(3.91)
2
2(3.91)
=7.81 �??????=ℎ
??????
���????????????�??????=??????(3.91)
2
(7.81)
=375.11 �??????
3
,�?????? 119.11?????? �??????
3

%=
98.75??????
119.11??????
=0.8291,�?????? 82.91% ��??????????????????????????????��
Optimizing Surface Area
98.75??????=????????????
2
ℎ
98.75
??????
2
=ℎ
??????=2????????????(
98.75
??????
2
)+2????????????
2
=
197.5??????
??????
+2????????????
2

�??????
�??????
=
−197.5??????
??????
2
+4????????????=0
4????????????=
197.5??????
??????
2

→ ??????
3
=
197.5??????
4??????



→ ??????=3.67 �??????
ℎ=
98.75
??????
2
=
98.75
(3.67)
2
=7.34 �??????
??????
���????????????�??????=2??????(3.67)(7.34)+2??????(3.67)
2
=253.69 �??????
2

8. Daniel Oliver, Matt Werno, Casey Burns, Rebecca Fenter

Red Bull Can: Graph: Volume vs Radius

Red Bull Can: Graph: Surface Area vs Radius

9. Daniel Oliver, Matt Werno, Casey Burns, Rebecca Fenter

Red Bull Can Analysis
Part 1:
The Red Bull can had a radius of 2.5 cm, a height of 15.8 cm, and a volume of 310.23 cm
3
. The
can was only 82.91% optimized. The can could have been designed to hold more volume by
increasing the radius to 3.91 cm and reducing the height of the can to 7.81 cm. This size can
would have the same surface area as the original, yet be able to hold 375.11 cm
3
.
Part 2:
The manufacturer did not make this can the cheapest way possible. They could have saved
money by reducing the height to only 7.34 cm and increasing the radius to 3.67cm. A can of
these dimensions would have a reduced surface area (and in turn, cost less to manufacture), yet
still hold the same volume as the current design. Since this can is a beverage, the manufacturer
likely chose to design the can for a better fit in a hand and not for either maximum volume or
minimum cost.

10. Daniel Oliver, Matt Werno, Casey Burns, Rebecca Fenter

Minestrone Soup Can: Graph: Volume vs Radius

Minestrone Soup Can: Graph: Surface Area vs Radius

11. Daniel Oliver, Matt Werno, Casey Burns, Rebecca Fenter

Minestrone Soup Can Analysis
Part 1:
The minestrone soup can had a radius of 4.15 cm, a height of 11 cm, and a volume of 595.17
cm
3
. The volume was 98.6% optimized. The can could have been designed to hold more
volume by increasing the radius to 4.58 cm and reducing the height of the can to 9.16 cm. These
dimensions would have the same surface area, but able to hold 603.64 cm
3
.
Part 2:
The manufacturer did not make this can the cheapest way possible. They could have saved
money by reducing the height to 9.12 cm and increasing the radius to 4.56 cm. A can of these
dimensions would have a reduced surface area yet still hold the same volume as the current
design. This would cost the manufacturer less money for each can.
Summary Analysis for All Cans
We noticed that with all of our cans that none of them were fully optimized for volume or
surface area (cost). It was easy to decipher why the Red Bull can was tall and thin because it was
made to be easily held in the average person’s hand. As for the cans of food, they were not fully
optimized due to the need to be able to be stored on shelves (cylindrical cans are easily stacked),
opened typically with a can opener (the larger the diameter, the more effort needed to open it),
and aesthetics (certain can shapes are more appealing, as well as certain brands rely on
familiarity for their marketing). There are other aspects to manufacturing to be considered as
well, such as: shipment crates and the ability to maximize the amount of cans per crate, cost of
more than raw materials such as welding costs or aluminum blanks, and processing and energy
costs of raw materials.