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I don't know how to solve for the smaller circle, but maybe we can first find the area of the blue region without the circle, that would be the area of the square - the circle, so the area of the square is (5*2)^2, since the radius is 5, the Diameter is 10 which is the side of the square, 100 is the area. then the circle is (pi)(r)^2, the radius is 5 so (pi)(25), 100-25pi, then divided by 4 because 4 regions is 25-25/4(pi), sorry idk how to solve the area of smaller circle but that's the idea, maybe Cphill or somebody wil give a splendid answer.

Ah.....the Golden Age of the Forum....I remember it well.....

Yep.....we have had some outstanding younger members  (DL included)

Merry Christmas to GA and all the others  on here that  make this thing somewhat legitimate (too numerous to name )

In honor of GA :

Note  the pattern :

99  = 1

88,89  = 2

77, 78, 79  = 3

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.

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11, 12, 13, 14, 15, 16, 17, 18, 19  =  9

The number of  two digit numbers  will just be the sum of the  first nine positive integers   =

(9)(10)/ 2  =   45

And we have  (99 - 10 + 1)  =   90  two digit numbers

P =   45/90      =  1/2

11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 44 , 45 , 46 , 47 , 48 , 49 , 55 , 56 , 57 , 58 , 59 , 66 , 67 , 68 , 69 , 77 , 78 , 79 , 88 , 89 , 99 , Total =  45 such numbers.

Probability = 45 / 90 = 1 / 2

It's a joke between Ginger and me.  There was a comedy duo called George Burns and Gracie Allen.  She played a wacky blonde.  He was the sedate, level-headed one.  At the end of their weekly show, they'd talk to the audience for a few moments, and they'd always close out by George telling Gracie to say goodnight.  He'd tell her "Say goodnight, Gracie."  She'd say "Goodnight, Gracie." 

_.

27/45     or  3/5

54 out of 90

28/45

I agree! and who is Gracie?

-Grace

Merry Christmas, Ginger.  Thank you for this wonderful post, and all your other posts.  I wish you contentment in the coming year, and all years after that.  Goodnight, Gracie.  ~Ron 

_.

Merry Christmas everyone! Thanks to all the great users of the forum that make the forum a better place!

Hope you alll have a great Christmas! 

- Grace

Let us simplify the guests answer, first expand the equation to get

5u+12u-18=-3u-27+20u+? = 17u-18=17u-27+?, subtract 17u so that -18=-27+?, add 27 to both sides so that

?=9, so \(9\)

The sum of the roots  =  a  + a^2  =  2

So  rearranging , we have

a^2 + a  - 2  =  0      factor

(a + 2) ( a - 1)  = 0

Setting each factor to  0  and solving for a produces  a =  -2    or  a  = 1

So  a^2  =  4      or   a^2    = 1

And the product of the roots = -k

So....the roots must  be   - 2  , 4   and    - k  = -8

So  k  = 8

Nvm it was 9

Call the intersections of the  medians, D.....this is the centroid

And  BD  =(2/3) BE = (2/3)(9)  = 6

And CF  =  (2/3)CF = (2/3) (12)  = 8

So....triangle DBC is a 6-8-10 right triangle  and its area =  (1/2) (6)(8)  =24

If we drew another median from A to BC....it would  divide this triangle into 2 triangles with equal areas.....so   the area of one of these  triangles  =12

And this triangle   = 1/6  area of ABC

So  [ ABC]  =  6 * 12    = 72

2 mod 6  =  2

We are looking for  integers n such that   4n  mod 6  =  2

n   =  2  5  8  11   14   17    20   23   26   29  32   35   38  41  44   47   50  53  56   59

20  integers

Thanks for the correction , jugoslav !!!!

See here  :   https://web2.0calc.com/questions/help_38214

Nope, it's not 7.

The distance between the centers is a hypotenuse of a triangle, and the legs are 1 and 7 units.

Distance is 5√2 units.

have fun on your run with the dogs. My dog just died...

*problem not answer

OMG I SOLVED IT BY MYSELF!!! 

btw i'm the one that posted the answer :) it was 11

i'm so happy!!!!

No, I'm just stuck on a workbook problem, and i'm bored since i finished all the other ones, and i want to learn from this one.

The hypotenuse  = 25

The   altitude length is    product of the legs  /hypotenuse = 15 *20 / 25 =  12

 The other leg  of  the right triangle in the  upper left of the figure  is 9

So...the area of this triangle  = (1/2) (product of the legs)  (1/2(9 *12)  = 54

And we can find the radius, R, of the inscribed circle as follows

54  =  (1/2) ( 9 + 12 + 15)  R

54 =  18  R

R  =  3

Similarly the  larger right triangle is a 12 - 16 - 20 right triangle

Its area  =  (1/2) (12 * 16)  =  96

And we can find the radius, R,  of the inscribed circle in this triangle as

96=  (1/2)  (12 + 16 + 20 ) R

96  = 24 R 

R = 4

The distance  between the  centers  = 3 + 4  =  7

INCORRECT>>>>>SEE JUGOSLAV'S ANSWER BELOW....

Sorry you need and answer so urgently..... I'll check in the morning to see if anyone has answered it....I'm going for a run with the dogs.

I'll pretend you are rolling 6   four sided dice

  say you roll 5   'ones'     there are 6 ways to do this

                      5   twos       6 ways

                      5  threes      6 ways

                       5 fours        6 ways

so 24 ways to roll   five-of- a kind

   now you can also roll 6 of a kind     four ways   :       four ones    four twos   four threes  or  four fours

so    24 ways to roll 5 of a kind    + 4 ways to roll 6 of a kind = 28 rolls

the number of possible rolls is 4⁶  = 4096

   28 / 4096 = 7/1024 = 0.0068359375                        (I think)