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are back in college? :| or you are taking a very weird job interview…
Just write down the equation to calculate the volume
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Neither. Surprising structures in reality are beautiful. *Why* is this the case? An equation is a poor explanation here.
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I get what you mean, but sometimes the equation answers the "why", when you know where different terms come from.
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yes, you have to start at from somewhere, it also depends on who are you trying to show it too… if it is a child…
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Is there a strictly geometric explanation? I only see the calculus answer, r ∝ sqrt(y) => A ∝ y => ∫A = 1/2V
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(But yeah, it's not obvious this would be exactly half. It freaked me out to learn a cone was exactly 1/3.)
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Scale so height is 1. Consider sweeping through horizontal slices z = h in [0, 1]. Ratio of radii /
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as function of height is sqrt(h), so ratio of disk areas is h, average ratio over [0,1] gives 1/2.
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