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Maths HL IA help using the brachistochrone model

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Im a hl maths student and i wanted to do my maths IA on a topic related to calculus as that was the option that my school choosed. And i came across optimisation of packagings. However ive seen that many people simply use the differentiation method to find the optimum surface area of the cans or products they are investigating. Hence to make it more complex i would want to use a unconventional theory or equation such as the brachistrone theory to find the optimum surface area of a can. Is it possible are there any other theories that i can use? and should i find the optimisation of a can or are there other things that i can find the optimisation of? or is this too simple, but what other complex and interesting calculus IA topics are there?

Edited by aditya124
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