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Math Help - two questions

  1. #1
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    two questions

    I don't really know if this goes here, so move it you have to. I have two sort of philosophical questions. first. I was thinking, if you cut lets say a coffee table in half, not straight down but from the side and then take that piece and flip it over and then cut that in half and flip it over, and so on. my question is can you technically cover infinite space like that if you keep cutting it in half? (assuming of course that we have a tool that can cut anything in half no matter what size). My second question is, lets say you're standing in one place and you are turning in a circle, what is the probability that you will shoot lets say a bullet at an exact point? for example, you start at some initial point at 0 degrees, and you are rotating about a fixed point, at any given time, you fire a bullet as you continually rotate, what would be the probability of shooting the bullet at an exact degree mark, like 207 or 15.3 degrees for example? Im thinking that there is an absolute chance that it will happen, even though there is a very very small chance it will happen.
    Can anybody try to explain this in any way?

    Thanks a lot
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  2. #2
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    Quote Originally Posted by nertil1 View Post
    I don't really know if this goes here, so move it you have to. I have two sort of philosophical questions. first. I was thinking, if you cut lets say a coffee table in half, not straight down but from the side and then take that piece and flip it over and then cut that in half and flip it over, and so on.
    The length of a table is measured by some postive real number, L.

    After 1 cuts you have,
    L/2

    After 2 cuts you have,
    L/4,

    After 3 cuts you have,
    L/8

    ...
    In general,...
    After n cuts,
    \frac{L}{2^n}.

    This number is always positive, thus you have a table for no matter what how small it is.

    The only thing you should realize is that,
    \frac{L}{2^n}\to 0 \mbox{ as }n\to +\infty
    Meaning the length of the table diminishes.
    (In fact at a collasal rate).
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  3. #3
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    Quote Originally Posted by nertil1 View Post
    for example, you start at some initial point at 0 degrees, and you are rotating about a fixed point, at any given time, you fire a bullet as you continually rotate, what would be the probability of shooting the bullet at an exact degree mark, like 207 or 15.3 degrees for example? Im thinking that there is an absolute chance that it will happen, even though there is a very very small chance it will happen.
    Can anybody try to explain this in any way?
    This is how I understand it.

    You are rotating on a circle.

    And you fire a bullet, and it does not travel straight because you are moving. So you want to know how it is effected by the rotating?

    If yes, the answer lies in something called "vector addition".
    The path of a fired bullet can be represented as a arrow which is the radius.
    The persons path can be represented as an arrow tangent to the circle because he is rotating.

    Below I attached a hand drawn diagram.

    The rectangle's diagnol is called vector addition it shows the result of the two velocities. If you know the speed of the man and the speed of the bullet you can calculate the resulting speed easily by Pythagorean theorem because it forms a right triangle. And also the angle by some trigonometry.
    Attached Thumbnails Attached Thumbnails two questions-picture11.gif  
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  4. #4
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    Quote Originally Posted by nertil1 View Post
    My second question is, lets say you're standing in one place and you are turning in a circle, what is the probability that you will shoot lets say a bullet at an exact point? for example, you start at some initial point at 0 degrees, and you are rotating about a fixed point, at any given time, you fire a bullet as you continually rotate, what would be the probability of shooting the bullet at an exact degree mark, like 207 or 15.3 degrees for example? Im thinking that there is an absolute chance that it will happen, even though there is a very very small chance it will happen.
    Can anybody try to explain this in any way?

    Thanks a lot
    The chance of firing at an exact angle prespecified is zero.

    Roughly speaking:
    Divide the full circle into N equal parts, the chance of firing in any
    one is p_N=1/N. Now as N->infty p_N ->0, so the chance of firing
    at an exact prespecified angle is zero.

    RonL
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  5. #5
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    but what about the bullet that you do shoot though? I mean its going to hit a specific angle, and what if that angle is the angle that is preferred? so what I'm asking is what are the chances that at that specific degree the bullet will hit the specific target you're aiming for. I know it will be very close to zero, but I don't think that it will be 0 if you repeated this experiment a lot of times.
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  6. #6
    Grand Panjandrum
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    Quote Originally Posted by nertil1 View Post
    but what about the bullet that you do shoot though? I mean its going to hit a specific angle, and what if that angle is the angle that is preferred? so what I'm asking is what are the chances that at that specific degree the bullet will hit the specific target you're aiming for. I know it will be very close to zero, but I don't think that it will be 0 if you repeated this experiment a lot of times.
    Lets suppose that the angle it does hit is well defined, the fact is the
    probability of having selected it beforehand is zero.

    A given angle constitutes a set of measure zero, while the totality of angles
    is a set of measure 360.

    Think in terms of selecting an interval I in angle space (so to speak) and
    consider the probability of hiting an angle in that interval. This is:

    p= L(I)/360,

    where L(I) is the length of the selected interval (assume this is <<360
    degrees if need be).

    RonL
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