Math Is Fun Forum

Well I just so happened to run into someone who explained to me the theory behind greenhouse gases and how they react, just in case anyone's interested. He wasn't sure about the supposed range in the atmosphere that the 'man-made' greenhouse gases were pooling at, but here's the general idea:

Most visible light comes streaming through the atmosphere and makes contact with the Earth. The Earth absorbs some of the energy and reflects the rest as infrared light.
Greenhouse gases are very good at aborbing infrared light and will do so for a portion of what is reflected off the Earth. When they absorb enough energy, they become active and then begin emitting the energy again. It radiates in all directions, so about half is radiated back up into space, and approximately the other half is radiated back at the Earth.

Still looking for a diagram of the altitudes that the different greenhouse gasses collect at, if anyone happens to know?

I'm still working on it myself. I agree with Johnny that it looks like it's 1/3rd, but I can't find an algebraic proof.

A few from one of my favorite authors, John Locke.

"Education begins the gentleman, but reading, good company and reflection must finish him."

"The improvement of understanding is for two ends: first, our own increase of knowledge; secondly, to enable us to deliver that knowledge to others."

"All men are liable to error; and most men are, in many points, by passion or interest, under temptation to it."

My personal favorite is:
"What worries you, masters you."

This is how I think about it:

When you do X/Y, you end up with X amounts of 1/Y. So in your examples, you have:
16 pieces of size 1/48
and
15 pieces of size 1/45

The fact that you have less 'pieces' of one is made up for by the fact that 1/45 is bigger than 1/48 so you don't need quite so many to acheive the same size.

I think a more extreme example would be: 5 pieces of 1/5 and 2 pieces of 1/2; they are equivalent, right? You can reduce 5/5 and 2/2 to 1/1, but not by the same factors (just like with 15/45 and 16/48).

Let me try a visual then to see if maybe I'm intepreting something wrong? Anyone's free to point out where I might be incorrect because now I'm worried that I've lost touch with Applied Maths.

B
56789
4
3
2
1
---------A

So the numbers here represent the individual links in the chain and the order in which they will hit the floor. A is the floor and B is the part where the links crest the edge. 

A) The acceleration exerted on links 1 to 5 is 10m/s² in the direction of A. It's gravitational acceleration, so it's a constant.
B) Links 1 to 5 have a positive vertical velocity in the direction of A, albeit they each have a different velocity.
C) Links 6 to 9 have a positive horizontal velocity in the direction of link 5, because they are being pulled along by the fall of the links before.
D.1) The exact moment link 6 is pulled over the edge of the table, it has 0 vertical velocity. The table will prevent it from gaining a vertical velocity until that point.
D.2) The exact moment link 6 is pulled over the edge of the table, it will experience the same gravitational acceleration (10m/s²) in the direction of A as all other links in the chain. (In order for it to have a greater acceleration, there must be something with acceleration >10m/s² acting on it, right?)

E) The velocity of an object is calculated as: initial velocity plus acceleration times the amount of time the object has experienced that acceleration, ie (V=U+AT).

So if the link that crests the table has had a total vertical acceleration in the direction of A of 0m/s² until that point, then how can it's velocity be that of the link at the bottom which has been accelerating at 10m/s² for an amount of time greater than 0s?

Identity has a very good point. Try out the hard ones as best you can and then check the solutions when you feel you're stuck or if you have an answer. It might put you in the right state of mind to deal with problems that are easier. There's some very good examples on there that get you thinking about how to turn a word problem into equations

Let's assume you don't pull down the first 10 meters (effectively connecting A to B with a length of chain).
So we start off the process and link 1 falls. Just as it touches the floor we freeze time so we can see what's going on and work out some equations.

At point A there is a link touching the ground that has fallen there from a height of 10 meters, beginning from rest. At point B there is now another link (the first link of the next 10 meters of chain) which is at rest, waiting to fall.

For the end of the chain (that has fallen to A) to start from rest and reach the floor 10 meters away, the time taken is determined by:
displacement = initial velocity * time + 0.5 * acceleration * time²
So:
10m = 0m/s * t + 0.5 * 10m/s² * t²
10 = 0 + 5 * t²
2 = t²
t = 1.414s (to 3 decimal places)

EDIT: Amended a mathematical mistake; subtracted 5 instead of dividing

So you now know that it takes 1.414 seconds for 10 meters of chain to fall from A to B. It will also take 1.414 seconds for the link at point B to fall to point A.

After the first 1.414 seconds, though, only the first link has hit the floor; there isn't any chain on the ground. For the first 10 meters of chain to be on the floor, the link at point B must hit the floor, right? Now because it also starts from rest, the time it takes to hit the floor is also 1.414 seconds.

For all 200 meters to be on the floor at point A, it will take 1.414s (for the first link to fall) and then 1.414s for each length of 10 meters of chain. So I make it 1.414 + 1.414*20, which is about 24 seconds.

Roel,

Absolutely right on the elevator one.

For this one:
1. The denominator of a certain fraction is 3 less than twice the numerator. If 5 is added to both the numerator and the denominator, the denominator becomes 1 more than the numerator. Find the original fraction..

Try writing out the statements in terms of equalities.

So:
The denominator of a certain fraction is 3 less than twice the numerator.
D = (2N)-3

If 5 is added to both the numerator and the denominator, the denominator becomes 1 more than the numerator.
D+5 = 1+(N+5)

The second one can be cleaned up. You don't need to add A+B+C in any specific order, so the parenthesis are unnecessary; I used them to better represent the sentence. So:
D+5 = 1+N+5

If you Subtract 5 from both sides, then it simplifies to: D = 1+N, as TheDude said. Because we know this, we can take the first equality and have only 1 variable in it. We do this by substitution:
1+N = 2N-3
Adding 3 to both sides:
4+N = 2N
Subtract 1N from both sides:
4 = N

Putting N back into D=(2N)-3, you work out that D does in fact equal 5, so your fraction is 4/5. Good job

As far as Mathsyperson's solution, he just did two steps at once. When he added it all up, it would have come out to:

If you multiply both sides by 12, then you will cancel out the divison and arrive at:

Good luck with your Law test. I have a friend who's studying Criminal Justice at the moment .

Sorry, Vibgyor, but I don't get what you're trying to describe or how you're arriving at your velocities? Can you try and explain it again and include some workings?

Not meaning to dig up an old thread, but I had an odd thought on the global warming...

Carbon Dioxide levels are normally what people attribute the rise of global mean temperatures to and are most worried about stopping, etc. They usually mention it being 'pumped into the atmosphere' which implies it sits like a bubble around the Earth, making everything underneath it hotter. You've all seen the pictures of the factory belching smoke into the air, I'm sure?

Carbon Dioxide is like... twice as heavy as nitrogen and oxygen, isn't it? And at least a little heavier than most other "atmospherically abundant" gasses, right? So isn't it really settling down nearer the surface of the Earth acting more like a blanket? If so, then aren't some elevations are suffering more than others; The Dead Sea would probably notice before Everest Base Camp would, wouldn't it?

Being completely honest, I don't read much on Global Warming anymore because I tend to find most articles arguing whether it's happening or not rather than explain what's actually happening. Pictures are welcome, but not required . A diorama would be nice; I haven't seen one of those in ages!

If you have two numbers, A and B, then the percentage of B that is represented by A is: (A/B)*100

If the price of 12 pencils is x, and 1x is the profit he makes from 8x, then the percentage percentage profit is:
(1x/8x)*100

By rules of multiplication, 1x/8x = (1/8)*(x/x) = (1/8)*(1) = 1/8

So, his percentage profit is: (1/8)*100 = (0.125)*100 = 12.5%

Sorry, Maths. Hadn't realised you'd replied.

Actually, I'm sure I can give you a short, but informative, description.

My High School covered me for Chemistry, Physics, (basic) Trig and Calc.
I then did a couple years which covered general Pure Math, Applied Math, Statistics, and some programming.
Absolutely nothing for another couple of years.
And now I'm in the middle of a math degree with probably a 40/60 split between Pure Math and Statistical Analysis.