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Venn diagram

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I think this one is a LEEEETLE  bit different thatn the others posted so far

1000 people    926   no military   so   74 military

out of 74 military    .567 are optimistic     74 * .567 = ~42 folks with military and optimistic

     ( 42 out of 1000 are + military and opt )

No military is 926     pessimistic is .582 of this    926(.582) = ~539 <=====  the rest are no military and optimistic = 926-539 = 387

SO total optimistic is  387+ 42 = 429   optimistic people

          42 out of 429  would be the optimistic with military hx = .0979       (I hope I didn't steer you wrong on this one !)

Venn diagram to follow...........

I think this one is a LEEEETLE  bit different thatn the others posted so far

1000 people    926   no military   so   74 military

out of 74 military    .567 are optimistic     74 * .567 = ~42 folks with military and optimistic

     ( 42 out of 1000 are + military and opt )

No military is 926     pessimistic is .582 of this    926(.582) = ~ 556 <=====  the rest are no military and optimistic = 926-556 = 370

SO total optimistic is   370 + 42 = 412   optimistic people

          42 out of 412    would be the optimistic with military hx = .102         (I hope I didn't steer you wrong on this one !)

Venn diagram to follow...........

Just use the Web2.0 calculator

 1.1421319796954315

  I suppose you have a graphing program somewhere that you are supposed to use

     here is what your graph should look like:

Draw a Venn diagram again

people with few program skills and a dog pref are outside of the diagram

1 - .118 -.052 -.224 = .606    

(Chris made a small error here:    Number feline pref only =  276 - 52  =  222 <------ should be 224   )

Just like the  other one, GM, let's suppose  that 1000  people were surveyed rather than just 100

Number (prog skills)  =170

Number ( feline pet pref)  = 276

Number (prog skills and feline pet pref )  = 52

So

Number prog skills only =  170 - 52 =  118

Number feline pref only =  276 - 52  =  222

Number   who  have few prog skills  and a canine pref  =

1000 - ( 118 + 52 + 222)   =  

1000 - 392  = 608

P ( few prog skills and canine pref)   =  608/1000  =   .608   =  60.8 %

Let the side length  of the larger hexagon  = S

And we can find the side length of  the  smaller hexagon as

sqrt [  2 (1/2 S)^2   - 2 ( 1/4 S^2) cos (120°) ]

sqrt [ 2 (1/2 * S)^2  -  2 (1/4)S^2  (-1/ 2) ]  =

sqrt [ (1/2)S^2  +  (1/ 4) S^2  ]   =

(S) sqrt [  3/4 ]   

These are similar figures so the ratio of  their  areas  = [ scale factor ] ^2  =  [ sqrt (3) / 2 ]^2  =  3/4

I think it is just a big counting problem.....  I count 32         anyone else ?

Find equation of one of the other lines in y = mx + b form

From the given line  -12/5 x + 2    the distance to the new line needs to be 8

   PERPINDICULAR line is   5/12 x + b

                      distance along  perpindicular line needs to be 8  from 0, 2

                           (x-0)^2 + (y-2)^2 = 8^2

                               use ratios to find the new point  (you should draw a picture)

                                       8/13 = x/12      x = 7.38462    from 0= 7.38462

                                        8/13 = y/5      y = 3.0769        from y =2   = 5.0769     This is our point on the parallel line

                                                                                                                                that is 8 units from 0,2

y= mx+b

5.0769 = -12/5 (7.38462)+ b     results in b = ~22.8     this is the y axis crossing    which is   20.8 from the original line y intercept

            the other parallel line  is also 20.8 from the original intercept      20.8 + 20.8 =

                                                                                                     41.6 units between the y-intercepts of the parallel lines     

The y intercept  of  the  given line =  2

The slope of this line =  (-12/5) 

A line perpendiculr to this one will have a slope of  5/12

And  this  line will pss through  the  point  ( 8 ( 12/13)  , 2 + 8(5/13) )  =  ( 96/13, 66/13)

So   the equation of  a parrallel  line  is

y =(-12/5)(x - 96/13) + 66/13

y = (-12/5)x +  1152/65  + 66/13

y = (-12/5)x  + 114/5

And because of  symmetry.....the distance  between the  y intercepts =  2 (114/5 -2)  = 2 ( 104/5)  = 208/5

See the graph here :

The  horizontal asymptote  is

y  = ratio  of [ coefficient on x^2 in numerator  / coefficient on x^2 in denominator  ] = 

y  = a   = 5 / 3

A mod 19 = 4
B mod 19 = 2
C mod 19 = 18, solve for A, B, C

A =19m + 4,  B=19m + 2,   C =19m  +  18, where m=0, 1, 2, 3.......etc.

Since A > B > C, then:

A=19*2 + 4 = 42
B =19*2 + 2 = 40
C =19*1 + 18= 37

[2A + B - C] =[2*42 + 40 - 37]==87 mod 19==11

Here's one way to  solve this....but....maybe not the fastest....!!!!

Let D  =(0,0)    E  =  (0, 1/2)    B  =(1.1)  C  = (1,0)

Call  the center of the circle  ( m, 1/2)

So  the distances from each of these points to the center  are equal  = the radius

We have this 

Distance from center to E   = distance from center to B

( m - 0)^2  + ( 1/2 - 1/2)^2  =  ( m - 1)^2 + ( 1/2 -1)^2

m^2  =  m^2 - 2m + 1  + 1/4

2m = 5/4

m = 5/8

So  the radius  =   the distance from (m,n)  to E  =  

sqrt  [ 0 - 5/8)^2  +  ( 1/2 - 1/2)^2 ] =

sqrt[  ( 5/8 )^2 ] =  

5/8

presently approx 412 parts per million

412 / 1 000 000  = .0412%

We must have that

A + B  =  6

So 

(A ,  B)

1     5

5     1

4     2

2     4

3     3

Note that if we  had a solid line, the points  (3, -4)  and (4, -3)  would lie on the  line

The slope of this line is  ( -3 - -4)  / ( 4  -3)  =  1/1  = 1

And the equation of this line  would  be   y = (1) ( x - 3) - 4  ⇒   y = x - 7

But....since we  have  a dashed line....we  have that  either

y < x -7       or  y  >  x -7

Sincs (0,0)  is in  hte  shaded area,  sub the  cooordinates into each equation to  see  which is true

0 < 0 - 7                0  > 0  - 7

0 <  -7                   0  >   - 7

And it's clear  that   y >  x - 7   is the correct answer

For simplicity....let's suppose that instead of  100 people  in the  survey, let 1000  people be in the  survey

Then  572  had trust in the government

And 319 had a widow's peak

And the number who had trust in gov and a widow's peak = 182

So...those  who only had trust in  gov  =572 - 182  = 390

And those that only had a widow's peak =  319 - 182  = 137

So....those  that  had no trust in gov and  no widow's peak  =

1000 -  ( 390 + 182 +137)   = 

1000 - 709   =  291

So....as the guest said...P ( no trust in gov and  no widow's peak) =  291 / 1000  =  .291

x^3 + y^3  =   ( x + y )  ( x^2 - xy + y^2)

And ( x + y) ^2   = 9^2   ⇒   x^2 + 2xy + y^2   = 81

And  since  xy  =  10  ....we have that

x^2 + 2(10) + y^2  = 81

x^2 + 20 + y^2   =81

x^2 + y^2  =  61

So

x^3 + y^3   =  ( x +y)  ( x^2 + y^2  - xy)  =  (9) ( 61  - 10)  =  9 * 51   =  459

Note that  x^2  + 7x  - 44  can be factored as  ( x + 11) ( x - 4)

So  we need  to  see if  x^2 + 5x + a    can be  factored using one - or both  - of  the factors above

Note  that if   a  = -36   we have  the factrization  ( x + 9) ( x - 4)

And the ratio of linear functions  is   ( x + 9)  / ( x + 11)

And  if a    =    - 66   we have the factorization  ( x + 11) ( x - 6)

And the ratio of linear  functions is  ( x - 6) / ( x - 4)

So  a =  -36   or a =  -66

And their  sum is   -102

2^n mod 13 = 3, solve for n

n = 12 m + 4 , where m =0, 1, 2, 3........etc.

The smallest n = 4  and  2^n =2^4 == 16.

So, the positive integers < 16 that are invertible modulo 16, which = k are:

1  -  The mmi =  1  [mmi stands for "modular multiplicative inverse"]
3  -  The mmi =  11
5  -  The mmi =  13
7  -  The mmi =  7
9  -  The mmi =  9
11  -  The mmi =  3
13  -  The mmi =  5
15  -  The mmi =  15
So, k == 8   and 8 mod 13==8