Question

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• a tall cylinder (like a smoothie cup)
• a cone - shaped paper cup
• a round glass sphere filled with jelly

they all appear to be about the same height - but which one holds the most?
the shop owner smiles and says, \theyre not filled the same way, even if they look close.\
questions:

1. what do you notice about the shapes and how much space they might hold?
2. what might affect how much each container can actually hold?

Explanation:

1. The cylinder has a larger volume compared to the cone (since the volume of a cone is \( \frac{1}{3} \) of a cylinder with the same base and height). The sphere's volume formula \( V=\frac{4}{3}\pi r^{3} \) and if we assume similar linear dimensions (e.g., height \( h = 2r \) for the sphere where \( r \) is radius, and for cylinder \( h = 2r \), base radius \( r \)), the cylinder \( V=\pi r^{2}h=\pi r^{2}(2r) = 2\pi r^{3} \), sphere \( V=\frac{4}{3}\pi r^{3}\). So cylinder holds more than sphere ( \( 2\pi r^{3}>\frac{4}{3}\pi r^{3}\)). So cylinder likely holds the most.
2. Factors like shape (cylinder, cone, sphere - different volume formulas), dimensions (radius, height for cylinder and cone; radius for sphere), thickness of material (e.g., if the paper cup has thick walls, inner volume is less), and whether they are filled to the brim (for cone, if it's a paper cup, maybe not filled to the very tip due to practicality) affect holding capacity.

Answer:

1. The cylinder likely holds the most as its volume formula ( \( V = \pi r^{2}h\)) gives a larger value than the cone (\( V=\frac{1}{3}\pi r^{2}h\)) and compared to sphere (assuming comparable linear dimensions).
2. Shape (volume formula differences), dimensions (radius, height), material thickness (affecting inner volume), and filling practicality (e.g., for cone) affect holding capacity.