#calculus project thing

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wide topaz
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Hello! I am currently studying for the IB Diploma and need assistance with my math internal assessment. My research question is: "Exploration of the Most Efficient Cookie Shape with Identical Volume for Packaging Purposes." I intend to use the concept of volume of revolution for this project, but I'm self-studying this topic and find it a bit challenging to grasp.

Here are the steps I have outlined so far:
The cylindrical cookie jar has a volume of 50 cm³ and a base area of 25 cm².
I plan to investigate various cookie shapes, including square, circular, star, and heart-shaped bases.
I want to use the volume of revolution to determine the area of each cookie shape. However, I'm unsure if this is the correct approach or if I need to establish specific values before starting my calculations.
I will multiply by the width to calculate the total volume used by each shape.
Finally, I will assess the remaining space in the jar to identify which cookie shape is the most optimal for packaging.

(basically what i want to ask is that whether volume of revolution is applicable here and what values i might have to establish)

royal flaxBOT
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modern cedar
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in addition to volumes of revolution

wide topaz
modern cedar
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well uh

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use all of the volume formulae to optimize them

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like for example with a cube

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V=s^3

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then find critical values

wide topaz
modern cedar
wide topaz
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or does this type of question not require that

modern cedar
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you could assume it is a cylinder

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but it’s really a volume of revolution of a modified sqrt(x) graph

wide topaz
# modern cedar but it’s really a volume of revolution of a modified sqrt(x) graph

wait so like:

  1. i set a range of value for x axis first so i can do intergration to tell what the volume of the cylinder is
  2. set dy/dx of V=s^3 to be 0 then find the maximum value of s
  3. then i multiply by the height of the cookie to measure the volume

but how can I calculate the leftover space that the cookies might have in the jar as its not possible to completely fill every nook and cranny of it? does that also involve calculus?

modern cedar
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find the height of the cookie to calculate the integral