Wednesday, April 21, 2010

Y!A: College Algebra Question

this is another question that i answered on Y!Answers, which can be found at this url:


Q:
John owns a hotdog stand. He has found that his profit is represented by the function p(x)=-x^2+66x+81, with p(x) being profits and x the number of hotdogs sold. How many hotdogs must he sell to earn the most profit.

my answer would be a million bajillion hot dogs lol. but my answer key says 33. help please

A:
haha yes, it would be nice if he could sell a million bajillion hot dogs, but look at your equation, do you notice that -x^2 term? that means that when he sells a lot of hot dogs, he'll be subtracting that huge number squared from his profits! that's really really bad for his profits!

also, if you'll notice, the formula for his profits is the equation of a parabola! not just any parabola, but one that is opening downwards. this means that there is maximum height that his parabola reaches, corresponding to his maximum profit, which makes this a maximization/minimization problem!

now, all that analysis aside, here's how you solve it:
since the slope (derivative, in other words) of his profits curve (the parabola) is zero at its maximim, we can take the dervative of the profits curve and set that equal to zero to solve the number of hot dogs:

P(x) = -x^2 +66x +81
so the derivative, P'(x) is
P'(x) = -2x +66
now we set that equal to zero (for the reasons mentioned above),
0 = -2x + 66
and solve for x (the number of hot dogs),
2x = 66
x= 33
voila!

to solve these problems in the future, try to analyze the problem as i did in the first two paragraphs. this will really help you get a grasp of the mathematics, and make solving these kinds of problems a breeze

Sunday, April 4, 2010

Y!A: BIGGER IS ELECTRICITY!

this is a question i answered for some guy on Y!Answers who had this CRAZY theory that gravity was somehow based on electromagnetism and the E&M forces were somehow giving rise to mass... and some weird other stuff. (i think he was trying desperately to unify E&M and gravity, but not succeeding very well! oh well, at least he's got GUTS!). but i answered his question to try to give him some idea of why that wouldn't work, and so i thought i'd post it.
[also note that at some point before i answered there were some other responses which he apparently got rather peeved about, and he responded in the "Additional details section". so that's what that is in case you were wondering. and i did not include the other responses, but his additional explanation was, at the least, interesting and relevant, so i included that.]

http://answers.yahoo.com/question/index;_ylt=Aqg4W49nJOBNV0qoLYrfJ8Pty6IX;_ylv=3?qid=20091019134800AAooDPq&show=7#profile-info-UMVkZmYOaa

Q:
How does gravity really work?
I just want to know if there is a DEFINITE theory or property that says how gravity works. We know it is there, we know certain properties, but how EXACTLY does it work. I have asked around a little, and nobody knows. I have my own little thought that i think is wrong, however if so, i want somebody to prove me wrong. What if gravity is actually a subset of the force of electromagnetism. We know for a fact that atoms have charges (or at least ions do) but even when they are neutral, they still have charges, they just balance each other out. What if these charges of the protons and electrons still individually have a field/charge of their own that is able to reach out to other atoms. Meaning, that if you have one pos magnet, and on neg magnet, they would attract. However, if you have 10 magnets trying to pick up 10 other magnets, the force is much greater. Same goes with gravity, the more mass/matter you have, the stronger the force of gravity. So my question is, can gravity be a subset or sub-something of electromagnetism? it seems to have similar properties to it, just molded and applied differently.

Additional Details
No, sorry, apparently, people didn't get what i'm saying. I didn't say that you need an electric charge for gravity to work, i said you need mass. How this mass has gravity? i said the electric charges FROM THE PROTONS and ELECTRONS that make up this mass have a charge, and these charges COULD induce gravity e.g. the more matter/mass you have, the stronger the gravitational/magnetic pull. And please don't say this is not possible just because there is another theory saying gravity is already this, that. Ignore those theories, or better yet, incorporate this one, and say if it is possible that gravity is caused by these charges at the molecular (and maybe smaller ) level. Basically can gravity be a subset of MAGNETISM, not electromagnetism, and don't say it is not because modern science says so.

A:
i think i understand what you are saying. here goes:

are you trying to just identify an analogy between electromagnetism and gravity, or are you trying to say that gravity is created by some charges (whether they are on a charged particle or even inside a neutral one, like a neutron)?

judging by your reaction to the first two posts, i'm going to guess that you don't mean the second.

if this is the case, you are right in your thinking when you said that "the more matter/mass you have, the stronger the gravitational/magnetic pull is". this is a good analogy. in electromagnetism, the more charge you have, the bigger the electric/magnetic pull is; similarly, in gravitation, the more matter/mass you have, the more pull there is. if this is all you are trying to identify, you are correct.

however, there is no correlation between how much charge there is inside the atom (or particle) and how much gravitational attraction it experiences.
for example, let's consider a proton and a neutron. they have almost exactly the same mass, but there is no charge on the neutron and there is a +1 charge on the proton. no matter how much charge is on these two particles, they will still exhibit exactly the same gravitational pull.
let's look at another example on the atomic scale.
let's say you have some heavy atom - let's say it's Krypton for now. Krypton has 36 protons, 36 electrons, and 48 neutrons (in one of it's most stable isotopes). we'll call this Kr-84 (for it's mass number). now let's also consider an ion of Kr with all of its electrons stripped away, which would also be Kr-84. if we were to put both of these atoms in a stable environment with one other insignificant mass, they would both feel exactly the same gravitational attraction to it (because the masses of both Kr atoms are the same).
however, consider a similar scenario. if we place both of these atoms again in a controlled environment with one electron, the Kr ion would feel a very large attractive force towards it. on the other hand, the Kr atom (with all 36 of its electrons) would feel no attractive or repulsive force (assuming it was far enough away such that the electrons on the outside of the Kr atom won't repel the electron much more than its protons will).

as you can see, there is no correlation between how much charge an atom (or particle) has and how much gravitational force it experiences. although your theory is very innovative and cleaver, we did not find that the gravitational attraction increased when the charge increased.

note that this is true for magnetic as well as electric charges, but it is just much more difficult to give an example for magnetism because you can never have an isolated magnetic charge.

i hope this answers your question about your theory of gravity! although it is not correct in this case, i would still write to einstein to see if he has any ideas on it :P!


but to answer you first question about the theory of gravity, we still do not know exactly how gravity acts and behaves. physicists today are working very vigorously on this problem and we have an idea of what it should be like.
first of all, there is a theory in physics (more specifically, particle physics) which states that every force is carried by a particle (this law is called the force-particle duality). in electromagnetism's case, it is the photon; the strong nuclear force is carried by the gluon; etc... . if you do not know much about modern particle physics, some of these ideas may seem very foreign and strange to you, but i'll give them to you anyway.
in the case of gravity, the supposed particle which carries the force is the graviton. although this particle has yet to be observed, most physicists agree that it exists, and have many complicated theories as to why we haven't observed it yet.
this being said, the gravitational force is still a mystery to most. the common theory is still einstein's theory of general relativity, which states that matter in space puts a curve in the fabric space-time (which we cannot visualize very well - actually, it is also extremely difficult to visualize the shape of space itself!). then, it is this curvature which brings bodies together under the force of gravity.

now, it is interesting to think about why more matter creates a bigger curvature in space. this is the theory of why matter has mass, and it is also related to the theory of gravity. this is what most physicists have their eyes on now and we are currently testing a theory based on the Higgs boson, which is supposed to be the particle which gives all matter mass.
you may not be very familiar with these concepts, but i'm giving them to you just in case you are!

this may not answer your question of a definite theory of gravity, but keep in mind that today's most brilliant physicists are almost as dumbfounded as the rest of us about the nature of gravity.

hope this helped though (at least the first part about your theory)! any other questions about what i said?