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Square root and IceBergs

Posted on 2002-06-11
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Hi

OK, two quetions for the price of one, what a bargain. :-)

Square Root
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Is there an actual mathematical formula for calculating the square root of a number ?? (not using tables)

Ice Bergs
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If the following are true...

1. Icebergs are made of Ice
2. Only the tip of the iceberg is visible (i.e. above the water)
3. Ice has a greater volume that water
4. We worry about global warming and the effect it will have on the sea level

... surely, if an iceberg melts due to global warming, the sea level would drop rather than rise, as the majority of the berg is under-water and therefore the volume of the entire iceberg would decrease
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LVL 55

Expert Comment

ID: 7070961
1, Only the Newton method of approximations as far as I know.

2, Eurika. a body that is lighter than water displaces it's own weight of water. The iceberg displaces the same weight and therefore the same volume of salt water whether it is frozen or melted or you carve the middle out and shape it like a big boat. Melting bergs will not make any change to the sea level, the ice that is currently on land as snowfields and glaciers however...
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Expert Comment

ID: 7071608
1- i know an interesting method and you can make a program to calculate it. Let's say you want to calculate the square root of 2. Now you take the x^2 - 2 equation and if you start to apply the Bolzano's Theorem to find the roots of the quadratic equation, you may aproximate the square root of two as much as you want. Another method would be using Series, but it will take me a while to explain ;).
2- I don't think i quite get it. More water makes the sea level rise. SO if the iceberg melts...
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LVL 101

Expert Comment

ID: 7071840
As andy says, an object in water displaces the same volume regadless of how it floats.  Think of that glass of ice water.  If you fill it with ice and water and let it melt the glass doesn't overflow when the ice melts.

mlmcc
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LVL 10

Author Comment

ID: 7072314
andylander, good explanation of '2'

So, if I fill a bucket with Ice cubes, top it up to the brim with water, and wait for it to melt, the bucket will remain totally full to the brim (excluding evaporation)

A friend of mine came up with another solution to this question, where he explained that because the water pressure was greater beneath the surface, the freezing point of water decreased, and therefore ice couldn't form. Which I would imagine in the case in the extreeme deep of the sea where the temperature is much lower than that of the surface ??? (but can you pressurise water)

mltolun, 'but it will take me a while to explain' - can you post example code ?? it doesn't have to be 100%accurate, just so I get the jist of it.

PS. Well done England
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LVL 17

Expert Comment

ID: 7073293
Nothing happened so I'll try again
Square roots

Here is a manual method example, itâ€™s similar to long division :

Pick a number, any number 18.6624

Draw a line above the number and a vertical line down the left hand side.

Starting at the decimal point, draw vertical lines grouping the numbers into pairs.

|---------------------
|18|.|66|24

Take the approx square root of the leftmost pair of digits 18, which is 4.

Write this number above the leftmost pair (add the decimal point if itâ€™s next)

4 .
|---------------------
|18|.|66|24

Write the square of this partial answer under the first pair and subtract.

4 .
|---------------------
|18|.|66|24
|16
| --
|  2

Bring down the next pair of digits

4 .
|---------------------
|18|.|66|24
|16
| --
|  266

Multiply the partial answer by 2 and write it next to the new line, with an underline to the right of it.

4 .
|---------------------
|18|.|66|24
|16
| --
8_  |  266

Find a digit (x) that will replace the underline so that (10 times the double the partial answer plus x) multiplied by x is less than the present remainder
or in this case so that x times eighty-x is less that 266.
4 . 3
|---------------------
|18|.|66|24
|16
| --
83  |  266
|  249
add this new digit to the partial answer on the top line and multiply the 82 by the new digit.

Subtract this new value from the line above and bring down the next pair of digits

4 . 3
|---------------------
|18|.|66|24
|16
| --
83  |  266
|  249
|------
|    1724

Again, multiply the partial answer by 2 and leave an underline to the right

4 . 3
|---------------------
|18|.|66|24
|16
| --
83  |  266
|  249
|------
86_  |    1724

Find a replacement for the underline similar to above, add this to the partial answer, and multiply

(the 862 by 2)

4 . 3 2
|---------------------
|18|.|66|24
|16
| --
83  |  266
|  249
|------
862  |    1724
|     1724
----------
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Subtract and continue until you have enough digits in the partial answer for your purposes.

In this case ( as I cheated) the result of the subtraction is zero so the answer is a proper square root.
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LVL 2

Expert Comment

ID: 7073817
you say in response to andyalder

"So, if I fill a bucket with Ice cubes, top it up to the brim with water, and wait for it to melt, the
bucket will remain totally full to the brim (excluding evaporation)"

BTW - you can't top it to the brim with water, the ice cubes will mostly overflow as they will float.

But why quibble - You misunderstood him; ice floats because water expands somewhat when freezing - ever leave a soda pop in the freezer and noticed how the can swells???
also much ice has air trapped in it too.

So... fill the bathtup half-full with water, fill a bucket with finely crushed ice, put the bucket in the tub, make a line where the water level is on the side of the tub, make a line on the bucket at the water's level on the bucket, come back in 8 hours, the ice is melted, the bucket is not full of water, the tub level is the same the displacement (bucket level) is the same.

considering computation of the sqrt of 2.  there is an interesting technique (what the heck!, it's a fascinating technique) called 'continued fractions', you can compute the sqrt of 2 as far out as you want.  I would give you the specifics here but my brain is rusty, I'll try to track it down.  BTW - I believe it is detailed out in a book "Mathematics in Everyday Things", William C. Vergara - I have found that book on the used book sites.  I lent that book to one of my students many (31 years) ago and the rascal never returned it.

chris-m

ice is slightly less dense than water, also most ice has some
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LVL 2

Expert Comment

ID: 7073827
a little bit hard reading, but try this for continued fractions.  http://www.mathpages.com/home/kmath434.htm
(my 'infinity friends' will note the author on that page wonders about what happens approaching infinity with this method).  chris-m
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LVL 55

Expert Comment

ID: 7073888
>"So, if I fill a bucket with Ice cubes, top it up to the brim with water, and wait for it to melt, the bucket will remain totally full to the brim (excluding evaporation)"

Wasn't my analogy but smegg's and I thought it was true up until now but I suddenly realise the level of water will be higher after the ice melts!!!

You can't have 2 seperate questions in one thread it gets too confusing so I'll post the cube in a bucket as a seperate question.
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LVL 55

Expert Comment

ID: 7076434
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Expert Comment

ID: 7081053
the sea levels would rise because the polar caps would break apart AND melt, having a double effect. the increased amount of water that was originally trapped at the poles, PLUS many many more icebergs floating away from the poles displacing water as well.
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LVL 55

Expert Comment

ID: 7084401
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LVL 1

Expert Comment

ID: 7088444
As others have mentioned, its the ice at the poles.
Specifically Antartica - It's not floating ice like
the north pole, but a large continent with 1500' of ice buildup.  Melt that, and we've got problems.
If I recall, if our average temp goes up (either 5 or 10)
degrees over the long term, we'll get 100' of sea level rise. That means the coastline shrinks back about 5-100 miles (depending on terrain) and we lose every coastal
city in the world that isn't on a cliff.

Now, whether we get that warming is open to debate...
but if happens, than the results have been calculated.

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LVL 3

Expert Comment

ID: 7090251
1. Sorry, couldn't resist the square root one (andyalder is correct). (Uses "Count" to prevent too many iterations):

Var  NO     Real, Var  Guess  Real, Var  Count  Ints
Start
;
;    Does the square root of an entered number
;
Print 'Uses the NEWTON-RAPHSON method to guess a square root.'
Prompt 'Enter number: ' : NO

Guess =  2.0 ; Starting point
Count = 0

While ((Guess*Guess) <> NO) & (Count < 10)
Guess = Guess - ( ( (Guess*Guess)-NO )/(2.0*Guess) )
Count = Count + 1
Wend

Print 'The square root of 'NO' is ' Guess

Stop

2. If the ice is all floating, no problem. If it's above sea level then ooops (again andyalder wins the prize...)
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LVL 55

Expert Comment

ID: 7090633
2. I was wrong. Since the berg is made from fresh water it displaces less than the equivalent volume of sea water so when it melts sea level rises.
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LVL 1

Expert Comment

ID: 7098321
I learned RobinD's technique in high school. Never used it though. It is more like a procedure than it is a formula.

You can estimate any function to any precision you want using a taylor series. Here are the first terms of the taylor series for the square root of x, around the point x=1:

x^(1/2) = 1 + (1/2)*(x-1) - (1/8)*(x-1)^2 + (3/36)*(x-1)^3 + ...

It is of course an infinite series. The more terms you consider, better is the precision. So you can calculate it to any precision you want.

==================================================

About the iceberg, the ice on land melting would raise the sea level. Also I proved that if the ice is fresh water and the sea is salt water, than the sea level would also rise.

check out:

http://www.experts-exchange.com/math_science/Q_20311344.html

But I really think that the floating icebergs would not contribute a lot to the sea level.
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LVL 1

Expert Comment

ID: 7098538
you are right... the floating icebergs would not contribute much to the sea level. Even at the north polar ice cap, the ice is not that thick. But the antartartic ice cap is very thick and over land. And it supports ice shelves that are 'suspended' over water for a great distance. (instead of floating with 10% above, they might have 15 or 20% above - I dont know the %)

floating icebergs and the arctic are irrelevent in terms of sea level rise
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LVL 5

Expert Comment

ID: 7102751
A suggestion on the square root question. A taylor series (or other similar series) would be one approach. This is how some early calculator chips did it.

Haven't got my tables book with me, but the details should be easy enough to find. I seem to recall that there is a method of working out how many terms are needed for a given accuracy.

Regards
Gordon
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LVL 1

Expert Comment

ID: 7103821
Gordon, check out my previous comment.
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Expert Comment

ID: 7109445
Oops

Gordon
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LVL 27

Expert Comment

ID: 7127403
I though that square roots were calcuated by the Newton-Raphson method? Better convergence.

If the tempurature rises too much Greenland will melt. There is two miles of thick ice on Greenland. If that melts we'll get another effect - the planet will spin differently (precission I think?). Mind you, I'll go and stay with the Mouse-on-Mars.
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LVL 1

Expert Comment

ID: 7130179
"Newton-Raphson method? Better convergence."

That is a numerical method. It give an approximate value to any precision you want. Check out:

http://www.sosmath.com/calculus/diff/der07/der07.html

"(precission I think?)"

Precession.
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LVL 55

Expert Comment

ID: 7130199
BigRat knows that. He's talking about Earth crust displacement, Hapgood/Einstein et al.
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LVL 27

Expert Comment

ID: 7131631
Rats can't spell. Voles are much better. They spend all day sitting in corn fields reading books. Yes, precession, is what I meant.

But what I actually mean is that the planet is not perfectly round (or ellipsoid for that matter) and that excessive mass in one place causes a change in the natural precession. If Greenland melts the water will distribute itself evenly over the planet relatively quickly, but it will take a great deal of time for the pressed in mass of the island to re-establish itself - and that will effect the precession and futher effect the climate.
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Expert Comment

ID: 7150150
"(but can you pressurise water)"
Nope. That's why hydraulics work.
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LVL 1

Expert Comment

ID: 7150222
If the "Can you pressurise water" comment is aimed at BigRat's comment about "pressed in mass of the island"..

The water isn't compressed. It's the land under the ice that's compressed. And the ice under the ice that's compressed. (You can't pressurise water, you CAN pressurize ice)
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Author Comment

ID: 7150387
sorry for being thick, but if water can't be pressuri(s/z)ed, why can't we dive deeper, surely this is due to the water pressure ??? or is that different ?
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LVL 27

Expert Comment

ID: 7153981
The original land of Greenland is under (I think) 2 miles of ice, and is pressed into the planet by about half a mile.

"why can't we dive deeper"

The weight of a substance causes pressure. Certain substances are compressible - ie: they get physically smaller under pressure. Water isn't one of them.

One cubic centimeter of water weighs one gram. So if you were under 1cm of water you'd have one gram on your head. Now go down 32,000 feet in the Mariana Trench. The weight of water on your head is now 32000 * 12 * 2.54 or 975360 grams or roughly one metric tonne.

HTH
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Expert Comment

ID: 7154053
...because air IS compressible and because your skeleton can't hold the weight of 1 tonne per square cm, the air in your lungs and middle ear gets a lot smaller in volume with the increased pressure. Hence you implode if you get too far down - a bit like sucking the air out of a drinks carton with a straw - it just buckles inwards. OK - reverse scenario, but you get the picute :)
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Expert Comment

ID: 7154055
Doh! ..picture.
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Expert Comment

ID: 7171434
NO, The melting ice won't change the sea level
YES, mschech is the only one to correctly point out that all the other posters were wrong!!  melting the ice caps wont change the sea level.
The threat from global warming is always mis-reported. There is not enough water in the pole to do anything, but there IS enough water in the oceans already to do something dangerous.  Its very simple - the oceans are like a bucket that is fill to the brim already.  A small thermal expansion will be enough to flood low-lying areas.

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LVL 55

Expert Comment

ID: 7171604
A melting iceberg in the sea WILL raise sea level. See the proofs in the question I linked to but basically because sea water is denser than fresh water and the iceberg is made of *fresh* water.
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Expert Comment

ID: 7175987
The density difference between ice and water is far greater than any density difference betwen fresh and saline, and as we have shown, that there is no effect from floating ice melting.
Suppose that after melting the fresh water layer does not mix very well, but continues to float on the salt water layer and therefore displaces it but a very small amount.
The eventual mixing of the layers removes even this displacement, so nothing gained.
The only possible effect you are left with is the conceivable volume change by mixing salt water and fresh, but that is negligible.  An reduction in average salinity of the ocean by melting bergs does not imply a volume change.
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Expert Comment

ID: 7175996
And anyway, the volume of water already in the oceans is far greater than any the poles.  Even a small thermal average expansion will flood the low lying areas. Above the lip of the bucket (the continental shelf) is the narrow spill zone that we call the shoreline.

The only interest in bergs and poles is that the indicate something is happening to the climate.
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Expert Comment

ID: 7177173
I believe the usual numbers thrown about are that if the WESTERN ice sheet melted we'd be looking at a 6-meter rise, of which much would be ocean thermal expansion.
The western sheet is less stable and is directly tied to ocean warming effects (it's in it). The eastern sheet is more stable, but far larger - if it melted we'd have a 60-meter rise, due directly to it's volume.
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Expert Comment

ID: 7177309
I love NASA's pun - a fragment of it's former shelf :) http://earthobservatory.nasa.gov/Study/LarsenIceShelf/

>and as we have shown, that there is no effect from floating ice melting.

On the contrary, we proved there *is* effect from floating ice melting but that there would be no effect if a salt water iceberg melted. Because icebergs are fresh the bucket overflowed.
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Expert Comment

ID: 7177846
>>we proved there *is* effect from floating ice melting

??

weight water displaced
= weight of frozen berg
= weight of melted berg.
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Accepted Solution

andyalder earned 20 total points
ID: 7178193
www.experts-exchange.com/math_science/Q_20311344.html

If it was a salt water iceberg (or a fresh water iceberg in a fresh water lake) then there would be no effect.  Because the 'berg is made of fresh water floating in a sea of brine it's initial displacement when frozen is less than the same displacement you would have if it was frozen brine but the volume taken up when it melts is the same since when you add salt to water it's density increases but it's volume does not.

Or in other words a million ton iceberg displaces the same whether it is formed from salt water or fresh but a million tons of salt water takes up less volume than a million tons of fresh water.
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Author Comment

ID: 8680278
Thanks everyone...

I thought I'd give you a year to agree / disagree !!!

Smg.
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