CPhill

15  8   14    7   6    8    2   8   9    3

The mean is 8

To find the variance...take the average of the squared differences from the mean

So.......we would have

(15 - 8)^2  +  (8 - 8)^2 + ( 7 - 8)^2 + .......+ (3 - 8)^2

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                              10

This ends up being    15. 2   =    [ population variance ]

Notice, EP, that all rotations of any arrangement are the same.....so.....a Repub assigned to any one seat can be in this same seat no matter the rotation....the only difference in the arrangements  are the number of ways that the other people can be seated around him  [ keeping in mind that we must have alternating seating patterns of R - D - R - D, etc ]

To see this more easily....in a simplified example....suppose that we have 4  people seated at a table

One arrangement is  

                                         1

                                 4            2

                                       3

This is the same arrangement [ with respect to rotations ] as 

                                          4

                                 3             1

                                        2    

So.....if we anchor  the first person at the " top "...the only difference in arrangements [ with respect to rotations ] are the ways the other three people are  seated around him.....note that another possible arrangement is

                                            1

                                    3             4

                                           2

If each member sits next to two members of the other party, they must just alternate D - R - D - R, etc

So...."anchor" one member, say, a Republican,  in any position

Going counter-clockwise  [ we could also go in the other direction ], we have

5 choices for the next seat [ one of 5 Dems ], 4 for the next [ one of 4 remaining Repubs ], 4 for the next [ one of the 4 remaining Dems, 3 for the next [ one of the 3 remaining Repubs ]....etc.

So...the number of possible arrangements  = 5! * 4!  =  120 *24  =  2880

 [ OAB]  =  30

And since AB = 10....then the altitude  of triangle  OAB  is

30  = (1/2) AB * altitude

60 = 10 * altitude

6 = altitude

And AB is a chord of the circumcircle.....and the altitude of OAB meets chord AB at right angles, so it bisects AB  ....so 1/2AB  = 5

And this altitude and 1/2 AB = 5   will form the legs of a right triangle with the circumradius as the hypotenuse

So

circumradius  =   sqrt  [ 6^2 + 5^2 ]   =  sqrt [ 36 + 25 ]  =  sqrt (61)

The median  of a right triangle drawn to the hypotenuse from the right angle has the same measure as 1/2 of the hypotenuse

We have a 3 - 4 - 5 right triangle wth BC  as the hypotenuse = 5

So   MB  = AM  =  (1/2) * 5   =   2.5

The diagonals of a rectangle are congruent and  bisect each other....

Distance between  (1, 5)  and  5, 3)  =

sqrt  [  ( 5 - 1)^2 + ( 5 - 3)^2 ]  =

sqrt [ 4^2 + 2^2]  =     sqrt [ 20]  

And if we connect these two points with a chord as well as drawing two radii to each point from the center of the circle, we will have an equilateral triangle....and since this triangle will have equal sides, the radii will have the same length as the chord

So....the area of the circle  =   pi * radius^2    =  pi * ( sqrt(20) ) ^2  =    20 pi units^2   ≈ 62.83 units^2

Angle AOB   = Angle MOC

And angle CMB  = angle ABM

So...by AA congruency....

Triangle BOA  is similar to triangle  MOC

And since  AB  =  AD = 4   and MC  = 2 .....then  the scale factor of triangle MOC to triangle BOA is 2 : 4  =    1 : 2

So

OC : OA   =   1 : 2   =   1 / 2 

Oops!!!....just a typo.....you are correct!!!

I have fixed it....

Angle BAC = angle ZAC  =  43°

Angle CZA = angle CZB = 90°

So triangle ZAC  is a right triangle and angle  ZCA  is complementary to angle ZAC  = 47°

And angle ZCA   =   angle HCA  =   47°

EDIT TO CORRECT AN ERROR>>>>!!!!

Find the area of the region bounded by the y-axis, the line y = 6, and the line y = 1/2x.

I have to go, Android.....

But.....the answer to this one is  36 units^2

I'll be back a little later to explain this one for you....but..at least you have an answer  !!!