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About Malle

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  1. This was entertaining. An additional observation is that when the column is flipped (23 seconds into the video), we can see the hour glass start to tilt. Specifically, the section with the sand falls down toward one side of the column while the empty section rises to the other side. This demonstrates that while the hour glass is net buoyant, the sand-filled section is not. Thus, when the column has finished flipping, there is a buoyant force on the hourglass, which results in a rotation as the sand filled section is displaced horizontally from the empty section, and only the empty section is net buoyant. This rotation gives rise to contact friction between the hour glass and the column, presumably sufficient to initially counteract the buoyant force and thus rendering the hour glass immobile. As the sand falls from the top section to the bottom section, the mass displacement between the two sections will be reduced. This leads to a reduced rotational effect, which leads to a reduced limit on the vertical component of the contact friction. When a sufficient amount of sand has moved, the buoyant force overcomes the friction, and the hourglass rises.
  2. Read it now, and I agree. You used a different bounding box, but otherwise it appears to be the same. Indeed. It is not quite a deadline as much as it is a time estimate. For what it is worth, you can define a more efficient algorithm if you are more prosaic about the order of dragon-space-points you search: Choose the point with the smallest distance to the origin which has not yet been excluded as a possibility. If there are multiple applicable points at the same distance, choose the one with the smallest angle from the x-axis, with angles measured in the interval [0,2pi)
  3. Yes. You want a constructive proof?
  4. No I don't, because the question didn't require me to show a constructive proof. EDIT: To clarify, you treat the "starting position" as the position of the dragon on the first day, not the position of the dragon when it started flying. If one were to provide a constructive proof, however, one could make use of (a slightly modified version of) .
  5. Solutions for 2 - 13 houses:
  6. Some easy to grab lower and upper bounds:
  7. Solution for the smallest case A simple argument for all larger solutions So all solutions, without explanation:
  8. References for definition of "average speed" or "average velocity": The original post states: It is not completely unambiguous for which time frame (for which total time) the car should achieve the listed average speed of 30 mph. A reasonable assumption is that it is from the start of the journey over the hill, due to the context of the question. We could also assume that it is from the start of the descent. We could also choose not to make an assumption about what the author wanted and instead parameterize the answer based on the time point from which we measure. This is left as an exercise for the reader.
  9. For what its worth, I did mean uncountably infinite. Typos. What I missed, clearly, is that A and S are countable, but not necessarily finite. Of course, from this we can create some sufficient (but not necessary) conditions for the problem:
  10. Am I missing something?
  11. You're welcome. The truth is that many things in geometry, algebra and calculus are strongly connected to modeling and understanding the world we live in or making calculations easier, but it isn't always easy to communicate that. Especially in pre-university classes there tends to be (in my experience) a lot of focus of how things work, but not why, because it is more important to get the pupils capable of answering the questions which will be on the tests, instead of teaching them the method in which the result can be found.

    I am not saying that it could be done the other way necessarily; it's likely more achievable to get the class to remember a few rules about the derivatives of a small set of functions than to teach them about the definition of a derivative, the epsilon-delta definition of a limit and how to handle limits with indeterminate form (where if you just substitute the value you get e.g. 0/0).

    But don't let that stop you from asking for more information. I don't know how good your math teacher is, but if they at least aren't so prideful that they cannot admit to now knowing an answer, then they should at least be able to point you in the right direction to finding more information. It may be easier to talk to them after class (when they don't have a large group of people to justify themselves in front of). It may also help if you ask them where to turn for more information, instead of asking them to explain it.

    And just remember, there's a lot of good sources on the Internet for learning math. Again, KhanAcademy has a lot of videos on a wide arrange of topics (not just math) and there are many others who upload lectures to YouTube. But sometimes you need the help to even know what to ask and in those cases turning to humans instead of search engines will definitely be of help.

  12. Wow, Malle you rock. Thank you so much.