In the “Canalysis” problem, we looked at the most used, and also found the ideal, sizes for soda cans. Once we found that out, we decided whether or not a common soda company, Coke, should switch to the more efficient can.
First, though, we had to actually find the most efficient can design. To do this, we found the volume, surface area, height, and radius of the original can size. Then, using the same volume (the same amount of soda), we made up two new cans. To do that, we came up with the height/radius, then solved for the other variable using the volume and variable we chose. After that, we found the surface area (in square centimeters), then multiplied it by the cost of aluminum per square centimeter.
First, though, we had to actually find the most efficient can design. To do this, we found the volume, surface area, height, and radius of the original can size. Then, using the same volume (the same amount of soda), we made up two new cans. To do that, we came up with the height/radius, then solved for the other variable using the volume and variable we chose. After that, we found the surface area (in square centimeters), then multiplied it by the cost of aluminum per square centimeter.
π22After we had a few cans each, we shared them out. First, we made the table of values (we used the radius and surface area), then we mapped them on a graph. Then, using the formulas we had needed previously, we made a function for the graph, and plugged it into a graphing calculator. The function came out to be SA=2πx^2 + 730/x. Once we had that, it was really easy to find the lowest surface area, which led to lowest cost.
The standard size for a Coke can has a height of 12.1 cm, and a radius of 3.1 cm. This led to a surface area of 296 cm^2, and a cost of $0.04736 per can. The best, most efficient size had a radius of 3.873 cm, a surface area of 282.7 cm^2, and a cost of $0.0452 per can. This may not look like a significant difference at first, but if you’re producing 19 billion cans a year, you save approximately $40,736,000 ($899,840,000 - $859,104,000). Does this make it worth it to switch, though?
I found that, no, it doesn’t. Coke would have to replace every vending machine, bottle holder, and piece of machinery that they used on the old size. Looking at how much money Coke makes every year, $40,736,000 doesn’t scratch the surface. The only pro to replacing the can would be the ability to market as “The New, Improved Coke Can!”. That wouldn’t produce nearly enough money to justify such a big change in the company’s manufacturing, selling, and buying.
The standard size for a Coke can has a height of 12.1 cm, and a radius of 3.1 cm. This led to a surface area of 296 cm^2, and a cost of $0.04736 per can. The best, most efficient size had a radius of 3.873 cm, a surface area of 282.7 cm^2, and a cost of $0.0452 per can. This may not look like a significant difference at first, but if you’re producing 19 billion cans a year, you save approximately $40,736,000 ($899,840,000 - $859,104,000). Does this make it worth it to switch, though?
I found that, no, it doesn’t. Coke would have to replace every vending machine, bottle holder, and piece of machinery that they used on the old size. Looking at how much money Coke makes every year, $40,736,000 doesn’t scratch the surface. The only pro to replacing the can would be the ability to market as “The New, Improved Coke Can!”. That wouldn’t produce nearly enough money to justify such a big change in the company’s manufacturing, selling, and buying.