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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,600

Mathegocart's thread The Earth and The String has this variation that is discussed from post #6 on.

The problem: The image shows my basketball (the earth, actually) suspended by a string that curves tightly around the base along a vertical equator up to the two tangent points, from where both sides continue as straight lines to the string's apex. Given that the string is 100 feet longer than the earth's equatorial circumference (radius 3959 miles), find the height of the string's apex above the top of the earth.

That string height is the sum of the length of the two tangents minus the length of the minor arc.

So...here's my Geogebra method. It also gave me some appreciation of the *enormous* scale involved!

1. Create point **A** at (0,0) - it's on the earth's circumference - and point **B** at (3959*5280,0) - it's the earth's centre, which you won't see just yet (the earth is big!)

2. In the Input bar enter "zoomout[3E6]" (without quotation marks, and the E must be uppercase)...you should now be able to see the centre of the earth.

3. Draw a circle (the earth) with the *Circle with Centre through Point* tool, centre **B** and point **A**, and move it closer to the bottom left corner of the Graphics window.

4. Draw a line through **B** perpendicular to the x-axis with the *Perpendicular Line* tool.

5. Create point **C** on the new perpendicular line, somewhere above the earth's top, with the *Point on Object* tool.

6. Draw tangent lines to the earth's circumference from **C** with the *Tangents* tool.

7. With the *Intersect* tool create the two {tangent,earth} intersection points (from left to right) and the {perpendicular line,earth's top} intersection point. They'll be points **D**, **E** and **F** respectively.

8. Hide the tangent lines (deselect *Show Object*).

9. Draw lines **CD**, **CE** and **CF** (in that order) with the *Interval between Two Points* tool (in some countries known as the *Segment between Two Points* tool).

10. With the *Circular Arc* tool select points **B**, **E** & **D** (in that order) to create arc **DE**. Colour it red.

11. Select Spreadsheet view, and make these cell entries (without quotation marks):

A1: "Tangents' length sum"

A2: "Minor arc length (d)"

A3: "Extra string (B1-B2)"

A4: "String height (CF)"

B1: "=i+j"

B2: "=d"

B3: "=B1-B2"

B4: "=k"

12. Set rounding to 15 decimal places in Options, and adjust the Algebra and Spreadsheet columns to display all text and digits.

You should now have something like this:

That was the easy bit...now for the hard part!

You've probably noticed that the spreadsheet's B3 value is miles over what it should be (which is 100 feet), but correcting that is trickier than I thought...

Adjust **C** vertically to &/or from **F** until B3 is as close to 100 as you can get it. For Geogebra to display its most accurate string height, B3 should be something less than 0.00000001 either side of 100, which can be achieved by zooming in & out manually and using functions *ZoomIn[<Scale Factor>]* and *CentreView[<Centre Point>]*.

Alternatively, enter "41807040Tan[th]-100-41807040th=0" into WolframAlpha, and plug the resulting *th* value (the fourth entry under *Numerical solutions*, which I increased to about 100 digits with the *More digits* option) into "20903520*Sec[th]-20903520" in WolframAlpha. Add the number from *Result* (no more than 12 decimal digits are needed) to the radius and replace **C**'s y value with the answer. B3 will update to a pretty accurate figure, confirming that our formula is correct.

And there you have it!

It's all rather fiddly. Surely there's a better way...

*Last edited by phrontister (2017-02-27 11:17:11)*

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Hi;

7. With the Intersect tool create the two {tangent,earth} intersection points (from left to right) and the {perpendicular line,earth's top} intersection point. They'll be points D, E and F respectively.

Got stopped right there. E = ? in the algebra pane.

Wait, I see you have B in a diiferent way than I do.

9. Draw lines CD, CE and CF (in that order) with the Interval between Two Points tool.

Now I am really stuck, what is the Interval between Two Points tool?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,600

You must have done something in a different order to my sequence to have that mix-up with E.

*Last edited by phrontister (2016-10-03 16:29:05)*

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,600

Now I am really stuck, what is the Interval between Two Points tool?

It's the second tool in the third-from-the-left toolbox in the Toolbar at the top of the page, in the same toolbox where the *Line* tool lives. I say "Line tool" hesitantly, which you'll understand if you read on...

On the Gebra site:

This tool has different names in different variants of English:

Interval between Two Points (Aus)

Segment between Two Points (UK + US)

Why do that? Clever!

I wonder what it's called in English-speaking countries other than Aus, UK & US?

*Last edited by phrontister (2016-10-03 16:24:53)*

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Hi;

Having a bit of a computer problem at the moment. Java is not running so The Geeb is not. Looks like that experimental kernel has done me a lot of damage.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,600

Ok...good luck!

*Last edited by phrontister (2016-10-05 00:02:49)*

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Hi;

Best I could do was 100.001958977198. I would suggest a slider for C.

Manually playing around with C using methods like interval bisection yields

99.9999999778812 and with that a value of 3888.61445999891

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,600

Hi Bobby;

The minimum slider increment is 0.00000001, so a slider on C can't get the accuracy. Pity the minimum can't be set closer to the max rounding. But at that level of accuracy, the length of the slider would have to be increased greatly to get the desired effect...you'd have to be zoomed right in before you could use it at that extreme level.

Anyway, I can now get Gebra's maximum (I think) string height accuracy of 3888.61446059 (for which B3 = 99.99999999988358) with my zooming routine in about 2 minutes, including using *zoomin[1000]* and centreview[C] three times at the end.

Gebra gives four more decimal points after the 9 (ie, 4952), but I've not been able to get it to show the next digit, 3, that is given by W|A, which has 3888.614460593728 (correct to 12 decimal places) as its result, working at 100-digit accuracy.

*Last edited by phrontister (2016-10-05 00:05:10)*

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Hi;

That is true about the slider.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,600

Yes, I think I'll let that thought slide...

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**Mathegocart****Member**- Registered: 2012-04-29
- Posts: 1,947

I am happy I could incite and spark debate about the enigma, the Earth and the String.

The integral of hope is reality.

May bobbym have a wonderful time in the pearly gates of heaven.

He will be sorely missed.

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,600

It's been very interesting...thanks for starting us off!

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

I have a M way to do the whole problem but there are a few steps that require some algebra. I was not able to solve the problem entirely using the new Geometry commands.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,600

So I guess that would be a different method from mine in post #21 of Mathegocart's thread.

I've since changed that to this, in radians:

```
r=3959*5280;
th=th/.FindRoot[2r Tan[th ]-100-2r*th==0,{th,0.02},WorkingPrecision->100]
h=r Sec[th]-r
0.01928719490295484482441479028250059270451052188967101077078489520191892182137176188340647018275855175
3888.614460593728239463379167602010692449187441461292338724354316092744378397068689244933991315452952
```

What answer do you get, and what is your M way?

*Last edited by phrontister (2016-10-06 22:25:15)*

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,600

bobbym wrote:

I was not able to solve the problem entirely using the new Geometry commands.

Do you mean the new ones in the recent new M version? I haven't upgraded...too costly and probably not needed for my junior use, I think.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

That is what I meant. Starting with 10.0, they added lots of stuff. I have only scratched the surface...

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,600

I have v10.2.0.0, and I'm still only starting out scratching the surface of *that*!

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

You scratch pretty good. With 28 year old claws it is possible. With 96 year old ones, it aint!

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,600

Even so, my scratching implements are much shorter and blunter than yours. My time is taken up with so many other things that M will always have to take a back seat. Can't see that ever changing...

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Can't see that ever changing...

Hard to predict the future.

Even this fellow could not:

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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