Fun With Math

[Ignatius J.]

I have an interesting problem from geometry. Well it came from the west wing, a show I really like.

Anyway, CJ turns to Toby and says "There's a spot on the earth where the temperature is exactly the same as it would be

if you drilled through the earth to the other side"

Will Bailey says to CJ:

"No, there isn't. "

So the question is: who is right? (and of course why?)

Edit: for clarification:

this is not a trick. Simply put, at any time are there two antipodal points on the earth with exactly the same temperature. The points do not neccesarilly stay at the same temperature, and at a later time, the points with the same temperature will be different, but for single snapshot of all the temperatures at the surface of the earth, are there two antipodal points with the same temperature?

[Barney B]

Originally posted by Ignatius J.

"There's a spot on the earth where the temperature is exactly the same as it would be

if you drilled through the earth to the other side"

Can we take it as a given that we should drill through the CENTER of the Earth?

EDIT:

Actually, a better way to say this is, "do the spots need to be antipodes?".

[dreamingwolf]

pick up a piece of earth and a drill a hole in it, its a riddle not math

also it could be a mound or a clump of dirt ect...

[MaddogCT]

It says "on earth" not piece of earth..so I'm going with the "Hole through the WORLD" perspective.......

Just a guess:

The North and South Poles?

:logo:

[dreamingwolf]

one can not guarentee the same temp on either side of the earth at the same time, the poles especially since they would be in different seasons.

Im not sure about ocean temps though, if it is going through the earth as a globe then thats probably the only way. But I still find it hard to believe that theres constant temps anywhere. But if there is, its probably ocean depths.

[Ancalagon the Black]

During either equinox, the angle of incidence of the sun’s rays would be exactly equal at 23 degrees North, x degrees West and 23 degrees South, (180-x) degrees East at about 6am at the northern point (6pm at the southern)...standard time, no daylight saving.

That doesn’t guarantee same temperature, of course.

[Ignatius J.]

No guys, you're thinking about it all wrong. Think about the temperature as a continuous function.

It has nothing to do with the fact that this is a temperature. If it holds for temperature it holds for any continuous function on a sphere.

And yes the problem is for drilling through the center. It's also not a riddle. The question is really just asking what you think it is asking.

[Ancalagon the Black]

Continuous and differentiable function?

[dreamingwolf]

Originally posted by Ignatius J.

No guys, you're thinking about it all wrong. Think about the temperature as a continuous function.

It has nothing to do with the fact that this is a temperature. If it holds for temperature it holds for any continuous function on a sphere.

And yes the problem is for drilling through the center. It's also not a riddle. The question is really just asking what you think it is asking.

oh I got it!!! if your glasses have a thickness greater than a coke bottle, you have the time to noodle it.

[Ancalagon the Black]

Originally posted by dreamingwolf

oh I got it!!! if your glasses have a thickness greater than a coke bottle, you have the time to noodle it.

Easy, cowboy. He did say it was a math problem in the first place, and you happily chimed in.

IJ, I don't see how the statement could necessarily be true.

[dreamingwolf]

Originally posted by Ancalagon the Black

Easy, cowboy. He did say it was a math problem in the first place, and you happily chimed in.

IJ, I don't see how the statement could necessarily be true.

I will forever hate you, you called me a cowboy.

[panel]

So the question is: who is right?

Answer: Will Bailey, cause the other statement is a bunch of BS that makes no sence.

The question is who is right, not how can you get the earth to be the same temp on both sides.

Either that or it has to do with you changing the earth because you actully drill a whole all the way across and that makes it work, maybe in the ocean or somethin that would have to do with that.

[Ignatius J.]

It has nothing to do with changing the earth in any way. Just a snapshot right now of the earth is there a point which is at the exact same temperature as the point across the center of the earth from it.

If it's not true try to come up with a counterexample. Constant temperature throughout the earth, it works, CJ's right. So try thinking up other ways you could have temperatures distributed on the earth. See if you can come up with a case that doesn't work.

Or of course, if CJ is right, then explain why.

And I'm not sure why it makes no sense, it's just a question of the earths temperature. It seems well posed?

Also, it is worth noting that I do not wear glasses.

[panel]

Originally posted by Ignatius J.

It has nothing to do with changing the earth in any way. Just a snapshot right now of the earth is there a point which is at the exact same temperature as the point across the center of the earth from it.

If it's not true try to come up with a counterexample. Constant temperature throughout the earth, it works, CJ's right. So try thinking up other ways you could have temperatures distributed on the earth. See if you can come up with a case that doesn't work.

Or of course, if CJ is right, then explain why.

And I'm not sure why it makes no sense, it's just a question of the earths temperature. It seems well posed?

Also, it is worth noting that I do not wear glasses.

The question makes sence, but the fact that there are to places exacly opposite of each other that can have the same temp. Air pressures, storms, cloud coverage, and different geography would surely insure that two places wouldn't have the same temperature all the time, even if they get the same % of sun light.

[Ignatius J.]

I'm talking about a snapshot, they're not always equal, just at any moment there are two places on opposite sides of the earth that are at the same temperature.

The answer is that CJ is right. Here's a hint:

Let f(x) be the temperature at a point x on the earth. Now, call the point opposite x: -x.

If you define a function g(x) = f(x) - f(-x), then the problem simplifies. In words, g(x) is the difference between the temperature of a point on the earth and the temperature opposite it.

Can anyone see where to go from here?

[dreamingwolf]

Originally posted by Ignatius J.

Also, it is worth noting that I do not wear glasses.

I was just being silly

[Leonard Washington]

Originally posted by Ignatius J.

I'm talking about a snapshot, they're not always equal, just at any moment there are two places on opposite sides of the earth that are at the same temperature.

The answer is that CJ is right. Here's a hint:

Let f(x) be the temperature at a point x on the earth. Now, call the point opposite x: -x.

If you define a function g(x) = f(x) - f(-x), then the problem simplifies. In words, g(x) is the difference between the temperature of a point on the earth and the temperature opposite it.

Can anyone see where to go from here?

i get it. so how is this funny?

i like the "rdrr" joke from the simpson's better.

[Joe Sick]

If you drilled through an anthill, wouldn't the temperature be the same on both sides?

It doesn't say "through the center of the earth."

Also, does the atmosphere count as being part of earth? Would seem that the temperature in space would be the same at high altitudes.

Really, I have no clue.

[dreamingwolf]

read the thread hes already clarified that

[herrmag]

I guess what you're saying is that at a certain time there may be an instance where exact opposite points of the globe are at equal temp's, yes? Not necessarily ONE specific area, but given your equation, at any given time it could be any two places on earth. Crap, I'm confusing myself. Write out the complete function. (sorry, by complete function, I mean with an example temp).

From what I understand, this just shows that if you plug in a number for x, use f(x) as a temp on one side of the earth, and f(-x) as a temp on the opposite, that there's a way to make g(x) = 0 [the difference in temp between the two points]. I really don't see how this proves that this occurs on earth, just that mathematically it is possible, but not that it ever has (or will) occur.

[Ignatius J.]

Yes, the points do not neccesarily stay the same temperature, just at any given instance there are such points. Later, those two points may not have the same temperature, but two other antipodal points do.

[herrmag]

Originally posted by Ignatius J.

Yes, the points do not neccesarily stay the same temperature, just at any given instance there are such points. Later, those two points may not have the same temperature, but two other antipodal points do.

So you're saying at all times this is occuring, but constantly changing in locations, correct?

[herrmag]

Oh, and btw, your title lied!!!! This is not fun, it's frustrating:laugh:

[Ignatius J.]

It's fun when you know the answer

and to your last post, exactly.

[DjTj]

Originally posted by Ignatius J.

I'm talking about a snapshot, they're not always equal, just at any moment there are two places on opposite sides of the earth that are at the same temperature.

The answer is that CJ is right. Here's a hint:

Let f(x) be the temperature at a point x on the earth. Now, call the point opposite x: -x.

If you define a function g(x) = f(x) - f(-x), then the problem simplifies. In words, g(x) is the difference between the temperature of a point on the earth and the temperature opposite it.

Can anyone see where to go from here?

Okay, I'll try some thinking out loud. I think I've forgotten all my math, and as I look above my desk, all I have is Black's Law Dictionary and Michie's Jurisprudence of Virginia and West Virginia, but let's try to keep this thread alive...

So we want to prove that there must exist an x such that g(x)=0.

We know it has to be continuous, so either:

(1) for all x, g(x)>0

(2) for all x, g(x)<0

or (3) there exists an x such that g(x)=0

if g(x)>0, then f(x)>f(-x) for all x, but if -(-x) = x, which should be true in our spherical geometry, then if you pick some x0, f(x0)>f(-x0) and f(-x0)>f(x0), which can't be true.

The same thing happens for g(x)<0.

So, there must be some x such that g(x)=0.

Does that work? It doesn't feel quite right, but like I said, all I have is law books right now.