Bafflers?

bobbym · 2014-02-26 22:06:11

Hi;

I got a formula like yours but had to settle for an approximation for the lune.

phrontister · 2014-02-26 22:46:19

If you have to use an approximation for the lune, does that mean you're eliminating different areas/shapes than I am? We must be doing something different, I suppose. Maybe you're not even eliminating anything and have some other method.

My numeric answer is exactly the same as stefy's (I checked to 30 decimal places), so I wonder how he went about it. His simplified formula is a bit different from mine, but I don't know enough about this kind of maths to be able to investigate and understand the difference.

Last edited by phrontister (2014-02-26 22:47:36)

bobbym · 2014-02-26 23:39:00

Hi;

phrontister · 2014-02-27 00:04:52

Oh. I've never done any calculus and know nothing about integrals, so I think I won't go there unless you can explain it to me fairly simply. Sorry.

Something else: I'd wondered how you gave that red shape its colour, but today discovered that you can layer in Geogebra. Is that how you did it?

bobbym · 2014-02-27 00:23:22

I did it even easier than that. I just copied the picture.

First can you see the small shape on the left. Do we agree that it is the same as the red region?

phrontister · 2014-02-27 00:32:09

Yes, I saw it earlier and from the symmetry of the whole construction can see they're congruent. That enabled me to divide by two in my calcs to eliminate the duplication and solve for the upper shape.

bobbym · 2014-02-27 00:34:35

You can get the shape on the left directly and then you know the red region too.

Give me some time to illustrate the process with geogebra.

phrontister · 2014-02-27 01:37:06

Sorry Bobby, but I'm going to have to turn in now. Had a verrrry late one last night, struggling with the lune until I got it to fit.

Catch you tomorrow!

bobbym · 2014-02-27 01:38:50

Okay, thanks for coming in. Tomorrow you can explain your layering technique.

phrontister · 2014-02-28 03:28:19

Hi Bobby,

There's a choice of ways to do some of this (and a lot of this you already know), but the following worked for me on my solution technique. It's pretty speedy too, once you get the hang of it (you'll take much longer to read this than to do it!)

1. Draw the square and circle like in your diagram, but make the arc a 'circular sector'.
2. Draw a 5x5 square in the N-E and the S-W corners of the square.
3. Set intersection points where the circle crosses the sector.
4. Right-click on any object and select Object Properties, then select the Colour tab from the Preferences window that opens.
5. Move the Preferences window right till about halfway off the screen to reveal most of the drawing, so you can see changes happen as you make them.
6. Select a conic from either the main screen's Algebra window (if open) or from the left column in the Preferences window.
7. Select a colour for it from the Preferences' palette and set its opacity to 100.
8. Do the same for the other conic and the 3 quadrilaterals, but with a different colour for each.

If the shapes now have the individual colours you intended, good; if not, still good, because that can be corrected via the Layer option on the Advanced tab on the Preferences window.

Layers are explained here, but just play with the numbers for each object (display hierarchy is 0 = lowest) and I think you'll soon see how it works.

Once you're done there, clean up the drawing, display the shape outlines (eg, in black), and do a final tidy-up:
1. Hide unwanted segment lines:
- in the Preferences window, left-click the 'Segment' heading in the left column;
- from the Basic tab deselect 'Show Object' (this also hides their labels).
2. Make all points and their labels visible and the same colour (eg, black):
- left-click the 'Point' heading to select all points;
- from the Colour tab select black;
- from the Advanced tab select Layer 9 (that's the highest layer, and lower layers can't hide higher layers);
3. Draw the large square, the circle and the 'circular arc' (previously the 'circular sector'), as all the shapes will have lost their black lines from the colouring done.
4. Draw a line from the large square's centre point to the following points on the large square:
- the midpoint of the bottom segment, the midpoint of the right segment and the bottom-right corner.
5. Hide conic and segment labels:
- open the Preferences window as before (it would have closed when you clicked on the main screen in step 3);
- left-click the 'Conic' heading and while holding down Ctrl select the 'Segment' heading, to select both groups);
- from the Basic tab deselect 'Show Label';
- close the Preferences window.
6. Final tidy-up:
- hide unwanted point labels;
- adjust the positions of remaining point labels;
- label the lines drawn in the large square's bottom-right quadrant.

That should just about do it, I reckon. If not, a small amount of tweaking should be all that's needed to fix it.

Or just holler if you end up with a mess you can't sort out.

Last edited by phrontister (2014-02-28 03:48:56)

bobbym · 2014-02-28 03:44:24

Hi;

Your method worked well, I did not even need the layering.

Why did you not create a new thread in Computer Math?

phrontister · 2014-02-28 05:42:29

Hi Bobby,

Yes, I didn't need layering either, which I didn't know when I first mentioned it. Back then I was adjusting an existing drawing where layering seemed to be the only way out, other than redrawing from scratch.

I didn't think of creating a new thread. Anyway, I'm a bit loath to open one because I'm confused about the colour-fill hierarchy of overlapping shapes that are on the same layer. From some experimentation I've done it appears that Geogebra's colour-fill hierarchy in relation to overlapping shapes that are on the same layer has some dependence on the order in which the shapes are drawn: ie, later-drawn shapes paint over earlier-drawn shapes. But not always, as I've discovered...and I don't know the reason for that.

I'm missing something, but what? However, I don't think it's important enough an issue to follow through; and really, I'm not too keen to spend more time on it than I have already, as layering is not something I'd use much at all...I think! So for the moment at least, trial and error and bumbling about (with a dose of persistence) will do me fine.

Btw, I looked at your new thread in Computer Math about this Bafflers problem and tried redrawing the image from it. So here it is...

bobbym · 2014-02-28 05:48:10

Hi;

Okay, thanks for the help on coloring it. I think geogebra should have a fill region command like most picture diting software has.

phrontister · 2014-02-28 11:08:47

Yes, that would be helpful.

When I started looking at your puzzle that is precisely what I expected to be able to do because I've done it with other editing software too, and I also expected that to enable me to somehow find the area of the filled region. I'd say that the latter would be unrealistic because of the effect of line thickness, though.

So a 'Fill Region' option would only be visual...but still quite useful.

bobbym · 2014-02-28 16:00:01

Geogebra can fill areas using the IntegralBetween or Integral command but it is more difficult than a simple fill command would be.

bobbym · 2014-03-06 07:32:51

New Problem:

From a deck of cards 3 cards are drawn successively with replacement. What is the probability of selecting 2 aces and a king?

Agnishom · 2014-03-06 13:38:24

'with replacement'? What does that mean?

bobbym · 2014-03-06 13:43:25

The card is picked then put back in the deck and reshuffled up.

Agnishom · 2014-03-06 13:48:20

Last edited by Agnishom (2014-03-06 13:57:46)

bobbym · 2014-03-06 13:56:28

Hi;

Please hide your answer.

Agnishom · 2014-03-06 14:18:44

???

Last edited by Agnishom (2014-03-06 14:27:52)

bobbym · 2014-03-06 14:22:02

Hi Agnishom;

Agnishom · 2014-03-06 14:23:52

yes, I edited post 1246, please check it again

bobbym · 2014-03-06 14:30:52

Hi;

Agnishom · 2014-03-06 14:31:49

Wait, I shall. Okay, hmmmmmm.

Math Is Fun Forum

#1226 2014-02-26 22:06:11

Re: Bafflers?

#1227 2014-02-26 22:46:19

Re: Bafflers?

#1228 2014-02-26 23:39:00

Re: Bafflers?

#1229 2014-02-27 00:04:52

Re: Bafflers?

#1230 2014-02-27 00:23:22

Re: Bafflers?

#1231 2014-02-27 00:32:09

Re: Bafflers?

#1232 2014-02-27 00:34:35

Re: Bafflers?

#1233 2014-02-27 01:37:06

Re: Bafflers?

#1234 2014-02-27 01:38:50

Re: Bafflers?

#1235 2014-02-28 03:28:19

Re: Bafflers?

#1236 2014-02-28 03:44:24

Re: Bafflers?

#1237 2014-02-28 05:42:29

Re: Bafflers?

#1238 2014-02-28 05:48:10

Re: Bafflers?

#1239 2014-02-28 11:08:47

Re: Bafflers?

#1240 2014-02-28 16:00:01

Re: Bafflers?

#1241 2014-03-06 07:32:51

Re: Bafflers?

#1242 2014-03-06 13:38:24

Re: Bafflers?

#1243 2014-03-06 13:43:25

Re: Bafflers?

#1244 2014-03-06 13:48:20

Re: Bafflers?

#1245 2014-03-06 13:56:28

Re: Bafflers?

#1246 2014-03-06 14:18:44

Re: Bafflers?

#1247 2014-03-06 14:22:02

Re: Bafflers?

#1248 2014-03-06 14:23:52

Re: Bafflers?

#1249 2014-03-06 14:30:52

Re: Bafflers?

#1250 2014-03-06 14:31:49

Re: Bafflers?

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