For this POW we began a segment on volume and how we can use a Coca-Cola can in order to help us with our knowledge on volume. We had to find the volume of an average Coca-Cola can and then create 3 more cans with the same volume (which isn't really the primary objective of the problem). We then had to find the surface area of of the can which we could use in order to find the cost of the aluminum (can). Once we found the cost, we all split up and wrote down everyone's values in order to put them on a line graph. Once we had all the data marked down on our sheet, we explored ways that we could re-write the height in terms of the radius to find a new volume formula. We then had to re-write the surface area in terms of the radius. We then looked back at the graph and decided on a radius that would produce the cheapest can (least surface area). Using the radius that we came up with, we had to decide what the height of the can would look like based on the radius. The final part of our Canalysis had us calculate how much money our new can would save.
For the process section of my Canalysis, I started by using the volume formula for a cylinder and found the volume of a standard Coca-Cola can. I then started plugging in values for the radius and height. I used values that created a similar volume but not the exact same volume. I shortly found that I could find the exact volume by doing the following: I used the volume that I found on the standard can and used it to create an equation in order to keep the volume the same on the can I am creating. I used the equation to create another can with the same volume. I then went on to surface area and cost of the can which was simple to do. I used the surface area formula (2π
r squared) + 2πr*h) and plugged in my values. I then multiplied the SA by $0.000016 because that is how much each square centimeter costs for aluminum. I then moved onto Act Two and gathered data with my classmates. Using the data I collected, I was able to work through the following: I found out how to re-write the height in terms of the radius by re-writing the problem so that the volume would be divided by πr(squared). For the SA, I started out by using the height formula I used on the problem before and then simplifying it appropriately so that it would work within the problem. Using the graph it was very easy to spot out which radius would result in the cheapest can to produce. Using the radius and volume that I acquired, I was able to find the height of the can. Now that I had the optimal height and radius, I calculated the cost of the standard can and optimal can, to see how much I would be saving.
ACT ONE:
Height12.1cm
15 cm
20 cm
Radius 3.1cm 28 cm 2.4 cm
Diameter 6.2 cm
7.84 cm 4.8 cm Volume 365.31cm(cubed) 369.45cm 361.91cm
Revised can with same volume (below) Height1.16
Radius 10 Diameter 20 Volume 365.31
S.A.296.06 cm (cubed)
313.15 cm
337.78 cm
Cost $0.0473$0.050
$0.054
For the process section of my Canalysis, I started by using the volume formula for a cylinder and found the volume of a standard Coca-Cola can. I then started plugging in values for the radius and height. I used values that created a similar volume but not the exact same volume. I shortly found that I could find the exact volume by doing the following: I used the volume that I found on the standard can and used it to create an equation in order to keep the volume the same on the can I am creating. I used the equation to create another can with the same volume. I then went on to surface area and cost of the can which was simple to do. I used the surface area formula (2π
r squared) + 2πr*h) and plugged in my values. I then multiplied the SA by $0.000016 because that is how much each square centimeter costs for aluminum. I then moved onto Act Two and gathered data with my classmates. Using the data I collected, I was able to work through the following: I found out how to re-write the height in terms of the radius by re-writing the problem so that the volume would be divided by πr(squared). For the SA, I started out by using the height formula I used on the problem before and then simplifying it appropriately so that it would work within the problem. Using the graph it was very easy to spot out which radius would result in the cheapest can to produce. Using the radius and volume that I acquired, I was able to find the height of the can. Now that I had the optimal height and radius, I calculated the cost of the standard can and optimal can, to see how much I would be saving.
ACT ONE:
Height12.1cm
15 cm
20 cm
Radius 3.1cm 28 cm 2.4 cm
Diameter 6.2 cm
7.84 cm 4.8 cm Volume 365.31cm(cubed) 369.45cm 361.91cm
Revised can with same volume (below) Height1.16
Radius 10 Diameter 20 Volume 365.31
S.A.296.06 cm (cubed)
313.15 cm
337.78 cm
Cost $0.0473$0.050
$0.054
ACT TWO:
a. h=v/πr(squared)
b. SA=2πr(squared)+2v/r
c. 3.8 cm
d. 8 and it would be slightly fatter but shorter.
Standard can: $8,930,000
Optimal can: $8,550,000
Savings: $38,000,000
I believe they don't change the can design of Coca-Cola because it is already so iconic and it would also take more money to change the machines, etc. Coca-Cola is already a very wealthy company and they don't really have to go through the trouble of designing a new can when they have so much money and it's already iconic in the world.
I learned how to manipulate formulas well in order to fit certain constraints definitely and overall organization. I believe I did well on this Canalysis and deserve a ten out of ten because I understood the problem even if I didn't at first (which happened). The habit of a mathematician I used for this problem was staying organized because it was really easy to get lost in this problem and forget exactly what you are looking for. I made sure to mark all my units correctly in order to not get lost and it paid off when I didn't mix anything up.
a. h=v/πr(squared)
b. SA=2πr(squared)+2v/r
c. 3.8 cm
d. 8 and it would be slightly fatter but shorter.
Standard can: $8,930,000
Optimal can: $8,550,000
Savings: $38,000,000
I believe they don't change the can design of Coca-Cola because it is already so iconic and it would also take more money to change the machines, etc. Coca-Cola is already a very wealthy company and they don't really have to go through the trouble of designing a new can when they have so much money and it's already iconic in the world.
I learned how to manipulate formulas well in order to fit certain constraints definitely and overall organization. I believe I did well on this Canalysis and deserve a ten out of ten because I understood the problem even if I didn't at first (which happened). The habit of a mathematician I used for this problem was staying organized because it was really easy to get lost in this problem and forget exactly what you are looking for. I made sure to mark all my units correctly in order to not get lost and it paid off when I didn't mix anything up.