CPhill

Let the number in the group  = 100

55 have  brown hair....so  (100 - 55) = 45  do NOT have brown hair

And  80 do not have red hair ...so(100 - 80)  = 20 DO  have red hair

So

Number who have red hair                                      20             4            

______________________________       =         ____  =     ___ 

Number who do not have brown hair                      45             9

The focal distance  is the distance between  the focus and the  vertex  = 1

This parabola turns upward because the  focus is a bove the  vertex

The form  is

4a ( y - k)   - (x - h)^2       where  (h,k)  is the vertex  = ( -1.3)

So we have that

4(1) ( y  - 3)  = ( x - - 1)^2

4 ( y - 3) = (x + 1)^2           divide both sides by 4

y  - 3  =(1/4) (x + 1)^2     add 3 to both sides

y = (1/4) (x + 1)^2  +  3

The  surface area of the cube  =  6s^2   (1)     where s is the side of the cube

And the  volume of the  cube = s^3

The surface area of the  sphere  =  4pi r^2    (2)   where r  is the radius of the sphere

Equating (1)  and (2) we have that

6s^2  = 4pi r^2          divide both sides by  4pi

6 s^2 / [ 4 pi ] =  r^2

[ 3 s^2 / (2 pi) ]  = r^2          take the positive root of both sides

r = s √ [3 / (2pi )  ]

So   the

Volume of Sphere              (4/3) pi r^3           (4/3) pi  [  s √ [3 / (2pi )  ]^3

_______________  =       __________  =    _______________________   =

Volume of Cube                    s^3                                s^3

(4/3)pi (3/ (2pi))^(3/2)  =

(4/3)pi * √ [ 3 /( 2pi) ]^3

                                        f(4)  - f(1)            244  -  484            -240

Avg rate of change  =     _________  =    ___________ =   _________  =     -80

                                           4  -  1                      3                        3

Note that 

w^3  = 1

w^3 - 1    = 0

(w - 1) (w^2 + w + 1)    = 0

So either   

w - 1  =  0      reject  because  w  cannot = 1       or

w^2 + w + 1  = 0

So.....

( 1 - w + w^2)  ( 1 + w - w^2)  =

[ ( 1 + w + w^2) - 2w ]  [ (1 + w  + w^2) - 2w^2 ]  =

[    ( 0)  - 2w ]  [ (0) - 2w^2 ]  =

[ -2w ] [ -2w^2 ]  =

4w^3  = 

4(1)   = 

4 

Remember that opposite angles in a parallelogram are equal

So  angle R  =  67°

And consecutive angles are supplementary

So

angle   QTS  + angle QST + angle RSQ   = 180

       67          +      63        +   angle RSQ   = 180

                   130                +  angle RSQ  =  180

                                            angle RSQ  =  180 - 130  = 50 °

And opposite sides are equal....so...

4x  =16

x  = 4

2, 3 or 4 girls must be selected

If 2 are selected we have  C(4,2) * C(8, 5)

If 3 are selected, we have C(4,3) * C(8, 4)

If 4 are selected, we have C(4,4) * C(8, 3)

So we have

C(4,2)*C(8,5)  + C(4,3) * C(8,4)  + C(4,4)* C(8,3)   =  672 ways

We can use the tangent-secant theorem to solve this

We have that   

y^2  =  7 ( 7 + 15)

y^2  =  7 (22)

y^2  =  154

And applying this once more, we have that

y^2  = 5 ( 5 + x)

154  = 25 + 5x     subtract 25 from both sides

129 = 5x             divideboth sides by  5

129/5   = x  = 25.8

The area  of (1/2)  the shaded  region inside the square is  =

Area of a quarter circle with a radius of 1cm  -  area of a right triangle with legs  of 1 cm  =

 pi (1)^2/ 4 -  (1/2)(1)^2  =

( 1/2)  [ pi/2 - 1]

So...usihg symmetry this shaded area  =    [ pi/2 - 1]

And the outer shaded area =

Area of a quarter circle with  a radius   of 2 cm   -  area of two quarter circes  with a radius  of 1 cm  - square with a side of 1 cm    =

pi * (2)^2 / 4  -   2* pi (1)^2 / 4  -  1  =

pi  -  (1/2)pi  -  1  =

(pi/2)- 1

So....the toal shaded area  =  

2 [ pi/2 - 1 ] =

(pi  - 2)   cm^2

Note that

When n = 1 to n = 3....the sum   = (1) *3  = 3

When n =  4 to n = 8, the sum  = (2)* 5 =  10

When n = 9 to n =15, the sum  = (3) * 7  =  21

So....generalizing this we have the sum

(1)*3 + (2)*5 + (3)*7  + ( 4)*9 +(5)*(11)  + (6)*13 + (7)*15  + (8)*17 + (9)*19 +  10   =  625