I had the opportunity to clear out my afternoon so I can visit the school I worked at last year.
Despite the difficulties I faced with job prospects (where I learned lots about life), that school (and especially my students) has a special place in my heart.
And so even within my busy schedule, I simply needed to make time to go visit my colleagues, and my students from last year. It felt great to see everyone again. It was awesome to see academic growth of the kids. I was also nosy enough to find out how my kids from last year are doing now.
I also bumped into some students that mentioned that they follow my blog. So here, guys, here's a special blog post for you all :) Thanks for the warm welcome, guys!
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When I made this blog, my intention was to allow myself to explore mathematics education, philosophy, and mathematics. I have not really explored mathematics yet. I have come across interesting questions... but since everything that I am doing is more focused on education, I have not had a chance to fully explore interesting questions.
So here it is, my first blogpost about a question that intrigues me!
Instead of outlining how I've tackled this question, I am just going to pose it and comment on it instead:
This picture above was actually just a random picture that "somewhat" relates to the question that I've been exploring.
So I was sitting around, and suddenly a question hit me. If I have a cone, how do I cut a circle into it so that what I've hollowed out has the same volume as the part remaining?
Let me elaborate by using a different way to explain this problem. Imagine you have a cake in a shape of a cone. Imagine you are cutting it with a circular cookie-cutter. Assuming you overlap the center of the cookie cutter with the center of the cone, how big does the cookie cutter have to be in order for the volume to be the same for the 2 pieces that you've removed? See below for a crappy visual that I made up:
Initially I thought this was a relatively easy problem. I posed the problem for other math teachers in my department to try -- and I haven't really gotten a good response on it yet. They haven't had a whole lot of time to tackle this, and they probably have other things on their plate, but it doesn't really diminish the fact that this interesting question is definitely more difficult than I initially thought.
so... what do you all think?
There are lots of extending questions that we can ask from this as well... I've also found a lot of related interesting results... but I'll save that for the next installment!
This is certainly an interesting scenario to tackle. The problem can be simplified in such a way that knowing the base radius of the cone or cake, the radius of the circle, or cookiecutter can be determined.
ReplyDeleteFrom the diagram above, the smaller cone can be called A and the truncated cone be called B. Using the volume of a cone and the fact that volume of A and B must equal, the following equation can be reached:
2(π/3)(R^2)(H)=(π/3)(r^2)(h)
where R and H are the radius and height of A, respectively.
Since cutting the original cone at a right angle takes us to similar triangles, H=(R*(h/r)), and substituting this into the equation creates:
2(π/3)(R^2)(R*(h/r))=(π/3)(r^2)(h)
2(R^3)(h)=(r^3)(h)
2R^3=r^3
Therefore, given r, the radius of the original cone, the radius of the circle or "cookiecutter" (R) would be:
R=((r^3)/2)^(1/3)