Determining the Optimum Shape of a Plastic Water Bottle

Introduction and Rationale:

With the topic of climate change and its increasing dangers, the environment is something which I have grown to become concerned about. Every day, there are new articles about the state of oceans, pollution, and its effects on the lives of animals. This could be from factors such as our growing population and the use of aluminum cans and plastics, which have increased substantially since the initiation of mass production in 1910. Due to the large use of plastics, the state of our environment has become the way it is today, which deeply concerns me (National Museum of American History, 2020).

As a result, I partake in service projects regularly such as picking up rubbish from a beach near my school while also taking part in ocean-based projects such as a 5-day sail on a traditional Fijian sailing boat with a crew who aims to raise awareness on climate change and its effects.

Initially, I was interested in determining the optimum shape of an aluminum Coke can.

This is because my favorite drink is Coke and I frequently purchase the can form. However, as I began to research the topic, I found that a larger percentage of aluminum cans are recycled as opposed to plastic bottles. Realizing that cans and plastics are deemed just as harmful as each other, I shifted my focus instead, to determine the optimum shape of a plastic water bottle. The reason for this is that plastic bottles are widely used around the world, with billions being sold annually.

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Due to large sales, it is difficult to maintain recycling processes for these plastics, with only 9% of all manufactured bottles being recycled (Nationalgeographic.com, 2020). In Fiji, plastic bottles are used almost every day by many, and therefore, determining the optimum shape of a water bottle would theoretically aid in reducing unnecessary wastage of these plastics, even if it’s by a minor percentage.

Aim:

This mathematical exploration aims to determine the optimum shape of a plastic water bottle to reduce the waste of plastics. For my exploration, I will be optimizing the plastic bottle shape of a local Fijian water brand ‘Aqua Safe’, to use the least amount of plastic possible. I wanted to use a ‘Fiji Water’ bottle at first but decided otherwise as I believed the Aqua Safe bottle would be more challenging as it is more curved. To find an answer, I will be using a fixed volume of 1500cm3 to calculate the surface area for different shapes. This will be done using calculus, specifics, integrals, differentiation, and optimization, to find an answer to the optimum shape (and dimensions) of a bottle to use the least amount of plastic. This means I will look for the shape which uses the least surface area.

Use of Mathematics:

To find an answer to my question, I decided to use calculus. I believed that what I learned in this topic could be easily applicable to finding the optimal water bottle shape. I felt as if using calculus could demonstrate what I had learned in the course and allow me to solidify my knowledge of differentiation and optimization, as I have always found it challenging. I also greatly enjoyed the topic, with this topic being one of my favorite topics within the syllabus.

To begin my exploration, I had to set certain parameters. I set the volume of my water bottle to one value, ensuring that volume is a constant variable and thus, making comparability easier. The volume I will use is 1500cm3(1500cm3 is equivalent to 1500ml). In addition, I realized that I would have to create some unrealistic conditions and assumptions to continue with the investigation. Firstly, I kept in mind that the actual volume of the bottle may be larger than1500cm3, as the bottle may not be filled to its maximum point. Along with this, in reality, bottles have caps and bottoms. However, I disregarded these two factors and instead, assumed that they added to the total volume of the bottle and its surface area.