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[MagmaMcFry]

Could you give an example?

[TheBiggerFish]

0(infinity) was stated, also infinity minus infinity, etc.

[Mostali]

I meant other indeterminate forms.

For type 0*infinity (f*g) then rewrite as f/(1/g) or g/(1/f).

For type 0^infinity or infinity^0 (f^g) then rewrite as e^((ln f)/(1/g))

For example:  Rewrite n^(1/n) as e^((ln n)/n)

For infinity-infinity you will need a common factor to rewrite as a product.  Then use the above if needed.

[Andres]

How do I convert surface area into volume? I have this dragon that has a surface area of 50m² but I don't know how to translate that into volume.

[crazysheep]

That question needs more information for us to proceed...

[TheDarkStar]

How do I convert surface area into volume? I have this dragon that has a surface area of 50m² but I don't know how to translate that into volume.

Are you asking how to get the volume if you know the surface area? If so, then yes, we need more information. Volume and surface area both depend on an object's shape as size, but not in quite the same way. Volume is a measure of the three-dimensional space while surface area is a measure of the outside surface of an object. They can also be changed somewhat independently - for example, if I took your dragon and flattened it into a sheet 1mm thick but didn't change the volume, the surface area would be very large - hundreds of square meters or more. If, on the other hand, I made the dragon into a sphere with the same volume, the surface area might only be a few tens of square meters. Without knowing some more details about the shape there's no way to get the surface area from the volume or vice versa.

There is some useful dependence, though. If you scale an object's dimensions up or down - in other words, if you make a second object that is similar but not congruent to the first - the square root of the surface area and the cube root of the volume of the new object will be the square root of the surface area and cube root of the volume of the old object times the scale factor.

[Andres]

Unfortunately, all I know is the surface area of the one dragon.

[Grek]

Unfortunately, all I know is the surface area of the one dragon.

Do you have a physical example of the dragon for analysis?

[crazysheep]

I think the more important question here is: what was the original problem that leads to the question of converting a surface area into a volume?

[Andres]

I think the more important question here is: what was the original problem that leads to the question of converting a surface area into a volume?

That's a secret, one that some fellow forumites will use to gain an advantage over me.

Ok, here's an update on the problem.
A human - by Dwarf Fortress standards - have a volume of 70,000cm³. The average surface area for an adult woman is 1.6m². The average height of an adult woman is 163cm.
A giant - basically an upsized human - has a volume of 9,000,000cm³.
What are the formulas used to translate volume into surface area and height? (I have different volumes to calculate and a formula is easier to put into a calculator.) Is this not enough information?

[Reelya]

Well if it's a literal upscale then you can deduce the scale factor by getting the two sizes. e.g. working forwards: If you scale by "F", then volume is F*F*F (F³) and surface area is F*F (F²). You can see this is true by considering a cube. If you double the scale of a cube, volume is x8 (it's now made up of 2x2x2 cubes of the original size) but surface area is x4 (each of the faces is twice as high and twice as wide as before).

So, to work out the scale factor "F" from two volumes, you divide the new volume by the old volume, to get relative volume, then take the third root of that, and that gives you the F factor.

e.g. for 70,000 to 9,000,0000 the volume increased by 128.5714 times. Take the third root of that, and you get 5.0471, which is the amount of scaling in each of the three dimensions. Since surface area is two-dimensional, that means and increase in surface area of around 25 times.

This is true mathematically, but in terms of physical engineering of structures or creatures, well a straight upscale doesn't work. The thickness of your leg dictates how much weight it can bear. If you double all proportions of a human, then the pounds per square inch that your leg is bearing doubles: the cross-section of your legs is x4 whereas your weight is x8. And this can't really be compensated for, because making a thicker body increases weight by the same amount that it increases load-bearing. The only way a true giant would work is by having short, stumpy legs and spindly arms, with the body tapering off to minimize upper body weight.

Unless you can come up with some radical new lightweight and strong bone material, you'd hit a point where your "giant" consisted of a pair of really stocky legs, and almost nothing else. Even in low gravity, there are other square/cube problems that break giants. e.g. blood pressure. Amount of blood grows a cube, whereas surface are of blood vessels and organs grows as a square. Meaning much more pressure on those tissues.

[Andres]

e.g. for 70,000 to 9,000,0000 the volume increased by 128.5714 times. Take the third root of that, and you get 5.0471, which is the amount of scaling in each of the three dimensions. Since surface area is two-dimensional, that means and increase in surface area of around 25 times.

So if I take the original human and increase its surface area by 36 times, I find the square root of that (6) and cube it (216) to get the final volume (15,120,000cm³). Is this correct?

So now that that's done, how do I calculate for the height?

[Andres]

Cool, thanks.

[jhxmt]

A problem that struck me while I was waiting in the incredibly slow queue to get coffee at work one morning, which I can't seem to get my head around - would be grateful for smarter mathematical minds than mine to give an opinion.

Scenario is: at work, I have a catering card.  I top up this card with money every so often, and then I can use it to buy things (i.e. coffee) from the coffee bar.

Assume my card starts with £0.00 on it.  Assume that, whenever I top up, I add £20.00 onto the card.  Assume that I only ever buy coffee on the card, and coffee costs £2.40.  Assume that if I do not have sufficient funds on the card to buy a coffee, I will add an additional £20.00 onto it.  I will continue this cycle until my card returns to exactly £0.00.  The card can never go into negative figures (so, if I have £2.39 on the card, I still need to top up with another £20.00 before I can buy a coffee).

Example: In this scenario, I would buy 8 coffees (leaving me £0.80 on the card), top up the card (so £20.80 on the card), buy another 8 coffees (leaves £1.60), top up (£21.60), buy 9 coffees and then finish (as card is now at £0.00).  So my total coffee consumption, with cost-of-coffee at £2.40 and top-up-amount at £20.00, is 8+8+9 = 25 coffees.

The hypothesis is that, for any pair of values* of cost-of-coffee and top-up-amount, the card will always return to £0.00 at some point, even if it results in me drinking a near-infinite amount of coffee.    Or, to put it another way, there is no set of figures that would result in an infinite amount of coffee.

Is there a straightforward way to prove or disprove this hypothesis?  It's been a few years since I stretched my mathematical muscles and I've bumped my head against this one a few times without making much progress.  I suspect I'm missing something incredibly simple and somehow being a complete idiot, but I just can't work out how to start proving/disproving the above.

* The values have to be legal financial values i.e. two decimal place figures.  I can't top up π pounds onto my card, nor can the coffee cost √2 pounds.   

[frostshotgg]

Case where it's not true: Card starts with any non-0 value already on it. Coffee cost = Refill value. Card will always be its starting value or starting value + refill value.