Canalysis
Problem Statement
Coca-Cola produces roughly 19 billion soda cans each year for the United States alone. Approximately how much money would they save by using the cheapest can instead of the current one ... and why do you think they don't? We were asked to experiment with different can sizes to help Coca-Cola out a little to see if they do save any money and if it is actually worth it.
Process/Solution
In order to figure out how much the company would save we needed to figure out the cost to make the cheapest can versus the cost of creating Coca-Colas standard can. We had to calculate the volume of the standard can first so that we could design the cheap can using the same volume.
(1) The height of the can was 12.1cm and the radius was 3.1cm, using the formula of volume for a cylinder (πr2h) the volume was 365.307cm3. I then experimented with different heights and radii. After coming up with these lengths I had to plug them into the volume formula to make sure the volume was the same as the standard volume.
(2) I figured out the areas along with the heights and radii and now it was time to figure out the cost. We were given that for one square centimeter aluminum costed $0.00016. Since the cost was calculated in square centimeters we had to calculate the surface area (2πr2 + 2πrh). Starting with the standard can, I calculated each cost by multiplying the surface area by the cost ($0.00016). The standard can costed Coca-Cola $0.047, my cheapest can with a height of 4.65 and a radius of 5 costed $0.049.
(3) After figuring out the cost of two other cans besides the standard can we needed some more information. As a class we decided to share our information with each other so that we could see who actually created the new cheapest can. After gathering multiple different can sizes from my class members, we created a graph to plot the points on (Radius = x, Surface Area = y). After plotting the points on an online graph we could came to a conclusion that the cheapest can should have the smallest surface area. On the graph we easily concluded that the lowest surface area was 131.81 with a radius of 2.64. The final part of this step was to create a couple of equations. The first equation I created was for the height but it had to be in terms of volume and radius. H= (V)/r2π. The second equation we had to come up with was the surface area in terms of radius. So I replaced the height (h) with the formula I got for height.
(4) Now that we had a lower radius and a lower surface area we could calculate the cost. The cost of the new and improved can came out to total at $0.0210896. The standard cost of the can was $0.047. So they would be saving some money and when they are producing 19 billion cans a year for the US alone they would be saving some money.
(5) If they are "saving some money" then why don't they switch cans? If we think about it in the long term and as a whole it would be completely worthless changing the shape of their cans. They would have to change all of their can making machines and come up with new packaging. Another reason they don't change their can shape is because the money that they would be saving is nothing compared to how much money they make a year. So as a conclusion it is completely point less to change the can shape.
Self Assessment
I feel like in this POW I learned about calculating volume and surface area and then also creating equations in terms of something (in this case radius). I also learned a surprising amount of information about Coca-Cola and just how much money they have... It is absolutely shocking. A habit of a mathematician I think I used in this POW was probably be generalize because when I solved each problem I was trying to understand how each problem was being solved in detail to make sure I understood it.
Coca-Cola produces roughly 19 billion soda cans each year for the United States alone. Approximately how much money would they save by using the cheapest can instead of the current one ... and why do you think they don't? We were asked to experiment with different can sizes to help Coca-Cola out a little to see if they do save any money and if it is actually worth it.
Process/Solution
In order to figure out how much the company would save we needed to figure out the cost to make the cheapest can versus the cost of creating Coca-Colas standard can. We had to calculate the volume of the standard can first so that we could design the cheap can using the same volume.
(1) The height of the can was 12.1cm and the radius was 3.1cm, using the formula of volume for a cylinder (πr2h) the volume was 365.307cm3. I then experimented with different heights and radii. After coming up with these lengths I had to plug them into the volume formula to make sure the volume was the same as the standard volume.
(2) I figured out the areas along with the heights and radii and now it was time to figure out the cost. We were given that for one square centimeter aluminum costed $0.00016. Since the cost was calculated in square centimeters we had to calculate the surface area (2πr2 + 2πrh). Starting with the standard can, I calculated each cost by multiplying the surface area by the cost ($0.00016). The standard can costed Coca-Cola $0.047, my cheapest can with a height of 4.65 and a radius of 5 costed $0.049.
(3) After figuring out the cost of two other cans besides the standard can we needed some more information. As a class we decided to share our information with each other so that we could see who actually created the new cheapest can. After gathering multiple different can sizes from my class members, we created a graph to plot the points on (Radius = x, Surface Area = y). After plotting the points on an online graph we could came to a conclusion that the cheapest can should have the smallest surface area. On the graph we easily concluded that the lowest surface area was 131.81 with a radius of 2.64. The final part of this step was to create a couple of equations. The first equation I created was for the height but it had to be in terms of volume and radius. H= (V)/r2π. The second equation we had to come up with was the surface area in terms of radius. So I replaced the height (h) with the formula I got for height.
(4) Now that we had a lower radius and a lower surface area we could calculate the cost. The cost of the new and improved can came out to total at $0.0210896. The standard cost of the can was $0.047. So they would be saving some money and when they are producing 19 billion cans a year for the US alone they would be saving some money.
(5) If they are "saving some money" then why don't they switch cans? If we think about it in the long term and as a whole it would be completely worthless changing the shape of their cans. They would have to change all of their can making machines and come up with new packaging. Another reason they don't change their can shape is because the money that they would be saving is nothing compared to how much money they make a year. So as a conclusion it is completely point less to change the can shape.
Self Assessment
I feel like in this POW I learned about calculating volume and surface area and then also creating equations in terms of something (in this case radius). I also learned a surprising amount of information about Coca-Cola and just how much money they have... It is absolutely shocking. A habit of a mathematician I think I used in this POW was probably be generalize because when I solved each problem I was trying to understand how each problem was being solved in detail to make sure I understood it.