CHRISTMAS CHALLENGE Question?

Question

Santa stands at x = 0 in an infinitely long and narrow corridor (defined by the x-axis). Initially the corridor is devoid of any stink particles. At time t = 0, Santa unleashes a stink bomb (ah political correctness) which releases N stink particles at x = 0. The particles move through the corridor by diffusion only.

Little Timmy’s nose is sensitive enough to smell anything as long as its CONCENTRATION is greater than C. Concentration is defined as the number of particles per unit x distance along the corridor. In terms of N and C, how far away should Timmy stand in order to never smell Santa’s stink bomb?
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FYI: mathematically, diffusion is governed by the following partial differential equation:
∂u/∂t = D* ∂²u/∂x²
-∞ < x < ∞, t > 0
where u = concentration of particles (units: particles / distance)
and D = diffusivity of the particles in the air (units: distance² / time)

10 points for the best answer, plus a load of thumbs ups. Ha ha, load.

What's the minimum distance Timmy should maintain?
Here's an idea of how the concentration varies with time and position.
http://tinypic.com/player.php?v=7w75enm&s=1

Nope. If C ≠ 0, the distance is very much finite.

There is no back of the book. This is a question I made up myself. But kaksi_guy and lewanj got it.

Answer 1

Solve

du/dt = D*d^2u/dx^2

with u(x,0) = N*delta(x)

The solution is

u=N*exp(-x^2/(4*D*t)) / (2*sqrt(Pi*D*t))

In order for Timmy not to smell the stink we need
u <= C

solving for x^2 we get
x^2 >= -4*ln(2*C*sqrt(Pi*D*t) / N)*D*t

The maximum of the right hand side occurs at
t = N^2 / (4*e*Pi*D*C^2)

substituting that into the inequality we find that Timmy is safe as long as
x^2 >= N^2 / (2*e*Pi*C^2)

which tells us that Timmy needs to stand at least N / ( sqrt(2*e*Pi) * C ) units away from stinky Santa.

Happy Holidays!!!

Answer 2

Given:

1. C = constant
2. Con. > C
3. Diffusion starts at x = 0

Using information:

1. As x increases, N decreases as x increases.
2. N/x = concentration

Using your equation:

1. What's that mean?
2. Well the varibles are given, so why not use them.

u > C
t > 0
D = rate diffusion occurs

3. Okay I think I got it.....

Here it is in a sentence:
The rate of change of the concentration with respect to time is equal to the rate of diffusion times rate of change of the rate of change concentration with respect to position.

Additional information found:

Concentration decreases as x increases,
Concentration decreases as time increases.

Here's the idea that people get wrong. If C is 0, it's guaranteed that Timmy will smell it, unless he's farther than what we can imagine. If C approaches 0, the same will happen. but what if C is a number that neither approaches 0 or is 0 itself? So we have to find it then.

More information found:

N/x > C

lim N/x -> C- ... This is related to the minimum distance Timmy will have to go.

If C is used as part of our answer, it's very likely that C is on the denominator.

So still, what is the equation used for?

What we can do with it:

1. Do nothing with it
2. USE IT!!!

I would select 2. if I could use it, but I can't. I have to do NOTHING with it until I find out how to use it.

To do this I must find an equation of u. u definitely can be written with x. How can we write u in terms of t?

As diffusion occurs, concentration will decrease. Diffusion moves as time increases. Using this we can find concentration in terms of t.
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.... I'm using my head too much. I'll hope this becomes useful to others.
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EDIT: I forgot to mention this. Bad Santa is a fake! I'm the real Santa. Bad Santa is the son of an unknown man in the streets.
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EDIT: Yea, it looks statistics is applied here.

Area under the curve is the same. Area under the curve = Integral f(x) dx. Also, isn't concentration suppose to be:
# of particles / AREA ?
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EDIT: Looks scary down there ( : ¹ O )

Answer 3

If you can tell me what time and where Timmy is when he is 1 standard deviation away from x=0 and he smells nothing, I will give you a big present.

Did I tell you that me wife has me on a strictly bean diet, and I might stink up the place when I slide down the chimney this year?

[If someone wants to cheat on the problem, Timmey needs to be at the inflection point which is one s.d. away]

Edit:

D.L. is from the good side of the family. I just deliver coal, stink bombs and flatulence. But my intuition on the answer is dead on -- you're going to max out where the normal curve inflects i.e (2*pi*e)^-2. Ha, Ha, Ha ..... No, Strike that .... Ho, Ho, Ho. And why do I say Ho, Ho, Ho -- well, I like them.

Answer 4

♠ oh, guys! It looks like heat conductivity and delta-function; well i try;

♠ oh, guys! It looks like heat conductivity and delta-function; well i try;
my handbook gives me a ready made solution:
u(x,t)=(1/√(pi*p)) *∫f(h)*exp(-(h -x)^2/p) *dh {for h=-∞ to +∞},
where p=4*D*t, u(x,0)=f(x) =N*δ(x);
♦ u(x,t)=(N/√(pi*p)) *∫δ(h)*exp(-(h -x)^2/p) *dh;
u(x,t)=(N/√(pi*p)) *exp(-x^2/p);
♣ now u(x,t) < C; (N/√(pi*p)) *exp(-x^2/p) exp(-x^2/p) <(C/N)√(pi*p);
x^2 > -p*ln[(C/N)√(pi*p)] =
= p*ln(N/(√pi*C)) – 0.5p*ln(p)= g(p);
♦ finding max g(p);
g’(p) = ln(N/(√pi*C)) –0.5– 0.5*ln(p) =0;
p= exp(2*ln(N/(√pi*C)) –1) =
= exp(ln((N/C)^2/(e*pi)) = (N/C)^2/(e*pi);
thus max g = -p*ln[(C/N)√(pi*p)] =
= -{(N/C)^2 /(e*pi)}*ln[(C/N)√(pi*{(N/C)^2 /(e*pi)})] =
= (N/C)^2 /(2e*pi);
thus x > (N/C)/√(2e*pi);
▬ as an engineer I should have guessed, if the answer is in meters then x must be proportional to N/C, so that I could take N=1, C=1 for a time being; the work out would look mmmmmuch easier! Be I always such clever as my wife is afterwards!

Answer 5

if he stands at minimum distance someone could push him into the radius of the bomb

Answer 6

infinitely to the MAX

Answer 7

Using separation of variables I get a general solution of u(x,t)=-1/2*exp(s4*t+s1)*
(2*s2
-exp(2*sqrt(s4/D)*(s3-x)))
*exp(-sqrt(s4/D)*(s3-x))/
sqrt(s4/D)
where s1,s2,s3 and s4 are constants. Maybe more later.

Answer 8

...he should stand in Santa's sack

Answer 9

same as above