SchrodingersCat.....

I must protest.

The earth has a diameter of 7917.5 miles; that is 41,804,400 feet. The surface area of the earth then would have 5.49E15 square feet of surface area. A really huge number. I didn't calculate the volume of this gold that would be 1.5 feet thick on the surface. BUT I think I can say without a doubt that the volume of the Earth's core is a lot less than the volume of this 1.5 foot layer on the earth's surface.

Do the math.

Even when he’s wrong, he’s right, because he’s made you a better man.

Who doesn’t dream of such a sentiment inscribed on one’s tombstone?

Doesn't work on my bank account or portfolio...sigh...

The difference in volume between a sphere of radius 20,902,200 feet and one of 20,902,201.5 feet is roughly 8.23541*10^15 cubic feet. That volume of gold would correspond to a sphere with a radius of about 125275 feet.

Obviously, all these numbers are overly precise.


Allowing a range of Earth radii between 3,950 miles and 3,963 miles, gives a volume of a 1.5-foot-deep gold covering which if reformed into a solid spherical core would have a range of radii of 125090 to 125364 feet.


So a core of solid gold at the center of the earth would have to have a 23.7 mile radius. The radius of the inner core is approximately 756 miles however.

Tyson



Edited 2 time(s). Last edit at 05/27/2021 12:01PM by Tyson Dunn.

OK, I'll do the math for you.

Formula for the surface area of a sphere: A = 4 * pi * radius^2
Formula for the volume of a sphere: V = 4/3 * pi * radius^3

The Earth's radius is about 4000 miles (approximation better than1%), which is 21120000 feet.

The area of the Earth is therefore about 5.6e15 square feet. The volume of the thin shell of gold (1.5 feet thick) is therefore about 8.4e15 cubic feet.

What size sphere would have that volume?

8.4e15 cubic feet = 4/3 * pi * radius_gold_sphere^3

So the radius of a sphere of gold with the same volume as the 1.5 foot thick shell is therefore about 126,000 feet or a little less than 24 miles.

So imagine that quantity of gold as a ball with diameter 48 miles, compared with the Earth's diameter of 8000 miles.

The Earth's inner core has a diameter of about 1500 miles and the outer core has a diameter of about 4200 miles. A sphere with diameter 48 miles would comprise a little less than two parts per million of the volume of the outer core.

Therefore, doing the math has shown that your hypothesis, about the volume of the Earth's core being a lot less than the volume of gold claimed, is incorrect.

Math is good.

CZ (who loves math almost as much as RfM)



Edited 1 time(s). Last edit at 05/27/2021 12:18PM by Concrete Zipper.

That's one of the wonderful things about math... and facts.

I did not go that last step. And of course math, the universal language, is correct. So I stand corrected.

Having spent a career working with very large or small numbers, I thought the 1.5 foot thick layer of gold was entirely plausible. That would be a very small percentage of the total volume of the earth. Thanks for the clear write-up of the math to confirm that.

Useful hack: if you have calculated to surface area of the planet in square feet, to calculate the volume of a 1.5 foot layer of gold, you don’t need to calculate two earth volumes, with and without the gold layer, and subtract them.

Simply take the surface area (5.6E15) and multiply it by 1.5. This treats the gold layer as if it were a flat sheet, but for a sheet only a foot and a half thick on a sphere of diameter 8,000 miles, the error is negligible, and it’s not like the other figures in this exercise are all that precise either.

“Close enough for government work.” ;)

Brother Of Jerry Wrote:
-------------------------------------------------------
> Useful hack: if you have calculated to surface
> area of the planet in square feet, to calculate
> the volume of a 1.5 foot layer of gold, you
> don’t need to calculate two earth volumes, with
> and without the gold layer, and subtract them.
>
> Simply take the surface area (5.6E15) and multiply
> it by 1.5. This treats the gold layer as if it
> were a flat sheet, but for a sheet only a foot and
> a half thick on a sphere of diameter 8,000 miles,
> the error is negligible, and it’s not like the
> other figures in this exercise are all that
> precise either.
>
> “Close enough for government work.” ;)

Close enough for almost any work, it turns out. Let's compute the exact answer and then the approximation, to show how close they are to each other.

The volume of a sphere is: V = 4/3 * pi * radius^3

Let's call the radius of the Earth 'R' and the thickness of the gold layer 'd'.

The volume of the inner sphere is: Vi = 4/3 * pi * R^3
The volume of the outer sphere is: Vo = 4/3 * pi * (R + d)^3 = 4/3 * pi * (R^3 + 3 * R^2 * d + 3 * R * d^2 + d^3)

Subtracting Vi from Vo we get the volume of the gold layer:

Vg = Vo - Vi = 4/3 * pi * (R^3 + 3 * R^2 * d + 3 * R * d^2 + d^3 - R^3)

Simplifying, we get:

Vg = 4/3 * pi * (3 * R^2 * d + 3 * R * d^2 + d^3)

And rearranging, we get:

Vg = 4 * pi * R^2 * d * (1 + d/R + 1/3 * (d/R)^2)

This is the *exact* answer for the difference of volumes of the two spheres. Now for the approximation!

If we assume that the thickness of gold is much less than the radius of the Earth (i.e. d << R), then d/R is much less than 1 and (d/R)^2 is waaay less than 1. This means that we can approximate (1 + d/R + 1/3 * (d/R)^2) as just 1, yielding:

Vg ~= 4 * pi * R^2 * d (where "~=" should be read as "approximately equal to"). This is the approximation that Brother of Jerry provided, where the area of the Earth is 4 * pi * R^2 and the thickness of the gold layer is d.

How good is the approximation? Let's try it with d = 1.5 feet and R = 4000 * 5280 feet = 21120000 feet:

d/R = 1.5 / 21120000 = 7.1e-8

1/3 * (d/R)^2 = 1.7e-15

Both of which are so much smaller than 1 that it's not worth it to add them in.

Using approximations like this one is important because otherwise, as DaveinTX pointed out, we are subtracting two very large numbers that are almost equal to each other. This is a numerically dangerous thing to do because small errors in the way those numbers are stored in a computer (or calculator) could have a large effect on the final result.

That's all the math for today. Why so much and why today? Because the past couple of days have been good for me, and happiness makes me erupt with math.

CZ (:-)



Edited 1 time(s). Last edit at 05/28/2021 12:21PM by Concrete Zipper.

And this is why lawyers and armchair philosophers should be careful pontificating about complex physics topics.

One is likely to lose his school money, proving that there is in fact such a thing as a free lunch—for others.

I am an engineer and a rock jock as well; now retired. I did the math all the way through to satisfy my curiosity.

Using 7917.5 miles for the diameter of the earth and 760 miles for the diameter of the inner core, I got the following with my trusty HP-15C calculator....

Volume of the Earth = 3.825292E22 Cubic feet.
Volume of the Earth with gold layer = 3.825292E22 cubic feet

The calculator display showed the same number, as it could not go beyond 6 significant digits in the display. But the DIFFERENCE between the numbers was 8.235408E15 cubic feet.

Volume of Inner Core = 3.383305E19 cubic feet.

So yes indeed math says the inner core is larger than this layer on the surface. What isn't taken into account is that Science says the Inner core is almost exclusively a liquid alloy of Iron and Nickel. But since the math says the inner core is 4654x larger than this layer on the surface it would not take a very high percentage of gold in the inner core to be able to make this gold layer.

I leave the geology to the geologists, like geologist Bernard Wood of Macquarie University in Australia, who calculated there's enough gold in Earth's core to coat its surface in 1.5 feet of the stuff.
I trust he knows WTF he’s talking about or his peers and Discover Magazine would have fact checked him and discredited him.

The really serious problem with 'trust' is that while competent to discuss the math he cited, is Dr. Wood 'trustworthy' in terms of his views on politics, religion, or diaper changing?

I have seen similar exercises where the calculations were off by a factor of a thousand, and sometimes off by a factor of a million, so it is worth checking the math. Brian Williams reported a figure live on air last year that was too large by a factor of a million. People noticed and you can google the incident. The problem is that Williams should have immediately recognized that the number could not possibly be correct.

I agree that the author, editors and readers of a publication like this would have caught gross errors.


BTW, I routinely see errors of two or four decimal places when people do percentages, and forget to move the decimal point two places, or worse, move it two places the wrong direction. :-/

Brother Of Jerry Wrote:
-------------------------------------------------------
> Brian Williams
> reported a figure live on air last year...

He never cleaned his calculator after that sandstorm

V ≈ 85,226.87 cubic miles

17,554 troy ounces per cubic foot.

How many cubic feet in 1 cubic mile? The answer is 147,197,952,000.

So: 147,197,952,000 x 85,227 = 12,545,239,855,104,000 cubic feet of pure gold in the 23.7 mile radius globe.

12,545,239,855,104,000 x 17,554 = 220,219,140,416,495,616,000 troy ounces of gold

Today's gold price: $1,901 (5/27/2021 @ 9:40am) times 220,219,140,416,495,616,000 = $418,636,585,931,758,166,016,000

... less your ten percent tithing ...

Except if there were that much gold sitting on the surface of the earth, the price would be zero, and you would have to pay people to haul it away.

On the other hand, a story about gold plates would be a lot more believable.

"The eventual intrusion of competent, grounded, repeatable data is what causes governments, religions, and love affairs to founder unless enough people are sucking that particular teat so as to make it inconvenient for Truth to become ascendant."

-- Agony DeMuerte, 27 BC to 27 AD (was year Zero a leap year?)

elderolddog Wrote:
-------------------------------------------------------
> (was year Zero a leap year?)

I know you're joking here, but I can address this topic seriously.

The Anno Domini system ( https://en.wikipedia.org/wiki/Anno_Domini ) that we use to number our years wasn't devised until the year A.D. 525, so years before then were never numbered as such. Also, even in proleptic (look it up) versions of the Anno Domini system, there is no year zero ( https://en.wikipedia.org/wiki/Year_zero ), so it would be difficult for a non-existent year to be a leap year.

That said, your question actually has a reasonable answer. Leap years in the original Julian calendar were applied incorrectly after that calendar was established, inserting an extra day every three years instead of four. To fix this problem, Augustus ordered that there would be no leap days added for a period of twelve years, from what we call 9 B.C. until after A.D. 4: https://en.wikipedia.org/wiki/Julian_calendar#Leap_year_error

Therefore, even if there had been a year zero, it could not have been a leap year.

You're welcome.

CZ

It's good to know stuff!

And it's a joy when people gift you with their knowledge.



There's an opposite to this, but I'll keep my mouth shut...for now.

Lol. I can now die happy.

Are you calculating the tithing on gross or net, EOD?

Depends on who's auditing the books, Dr. Moneybags . . .

I'm not exactly Dr. Moneybags. I earn enough to pay the bills without a great deal to spare. My wife (or my kids) may be quite wealthy someday.

Then you must have some very luxurious bills!

The property taxes here are outrageous.

Does gold float on the surface of water?


I.Don't.Think.So

Gold can and frequently does float on water.