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Hi

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

Square Root

-----------

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

Ice Bergs

---------

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

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

Square Root

-----------

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

Ice Bergs

---------

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

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...

2- I don't think i quite get it. More water makes the sea level rise. SO if the iceberg melts...

mlmcc

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

Square roots

Here is a manual method example, it’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 it’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

----------

0

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.

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

(my 'infinity friends' will note the author on that page wonders about what happens approaching infinity with this method). chris-m

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.

Here is an iceberg, http://news.bbc.co.uk/hi/english/sci/tech/newsid_1731000/1731163.stm

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.

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...)

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.

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

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

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.

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.

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.

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)

"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

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.

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.

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

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.

>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.

??

weight water displaced

= weight of frozen berg

= weight of melted berg.

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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.