CPhill

Note  that   f (-2)   =  6  and that f(2)  = 6

Which means that  f(x)  =  -2   or  f(x)  = 2

Note that  there is no point on the graph  where  f(x)  = -2    becaise  the graph is never negative

So....it must mean that  we are looking for   f (x)  = 2

Note that there  are three places on the graph where  f(x)   = 2   

There is actually  an algebraic way to show that there are three values which give us   f (f(x))  =  6

The slope of  the leftmost segment of the line  = (6 - 0) /(-2 - -4)  =  6/ 2  =  3

The equation of this segment  is   y = 3 ( x - - 4)  =  y = 3x + 12

So....we are looking  for the  x value where     f(x)  =2....so ...

  2 = 3x + 12  ...so   -10  = 3x  and x = -10/3

So    f (-10/3)  =  2

So   f ( f(x))  = f ( f(-10/3))  = f (2)  = 6

And the slope of the middle segment on the  graph   = (6-0)/(-2-0)  = -6/2  = -3

And the equation of this segment is  y = -3x

So....we are looking for the  x value where  f(x)  = 2  ...so

2  =  -3x ...  so     x = -2/3

So   f( f(x))  = f ( f(-2/3))  = f (2)  = 6

Finally......the slope of the leftmost segment is the same as the leftmost segment = 3

And the equation of this segment   is  y  =3x

So.....we are looking for the  x value  where f(x)  = 2....so

2 = 3x   ...so    x  = 2/3

So  f ( f(x))  = f(f(2/3) =  f(2)  = 6

And this confirms that   there are three solutions for  f (f(x))  =  6

I just decided to take a little  (well,,,,really....a long) break, Eternity.....good to see you guys, again !!!!

The two equal sides must  = 20.......here's why.....by the triangle inequality,  the sum of any two sides of a triangle must be greater than the remaining side....if the two equal sides   =10, then 10 + 10  =  20  which is NOT greater then the remaining side (20)

So....the sides must  be 20, 20  and 10

We can use something known as Heron's Formula to  solve this

First......sum the sides and divide by two  =  (20 + 20 + 10)  / 2  =  25

This is known as the semi-perimeter, S

The area  =   sqrt  [  S * (S -A) * (S -B) * (S-C)  ]    where A, B ,C  are the sides of the triangle

So we have

Area   =sqrt [  (25 * (25  -20) (25 -20) (25 -10) ] = sqrt  [ (25) * (5) (5) (15)] 

Area  =  sqrt  [ 25 * 5 * 5 * 15 ] =

Area  =sqrt [ 9375 ]  ≈  96.8  sq inches

See the following  image  :

Let O = (0,0)

Let A = (2,0)

Let B = (0,2)

Let E = (-2,0)

C =(0,1)

Note that BE  is the diameter  = 4

And  CE  = BE - BC  = 4 - 1  =   3

Since AOB is right  with OA =  2   and  OC =  1

Then, by the Pythagorean Theorem  AC  =sqrt  ( OA^2  + OC^2)  = sqrt  ( 2^2 + 1^2)  =  sqrt (5)

By the intersecting chord theorem  we  have that

BC * CE  =  AC * CD

1 * 3  =  sqrt (5) * CD

3 = sqrt (5) * CD

CD =  3/sqrt (5)  ≈  1.34

THX!!!!!

A = (3,6)     B  = (-5,2)    C  = (7, -8)

Midpoint of  BC = [ (7 - 5) / 2 , (-8 + 2)  / 2)   = ( 2/2   , -6/2)   = (1, -3)  = D

Slope  of  line through  AD  = [ 6- - 3 ]  /  [ 3 - 1 ] =  9/2

Equation of line   through AD  ...   y = (9/2) ( x -3) + 6  ......y = (9/2) x - 27/2 + 6  ......y = (9/2)x  - 15/2

Midpoint of  AC =  [ (3 + 7)/2  , (-8 + 6))/2  ]  =  (10/2, -2/2)  =  ( 5, -1)  =  E

Slope of line through  BE  =  [ -1 - 2 ] / [ 5 - -5]  =  -3/10

Equation of line  through  BE....y=  (-3/10) ( x-5) - 1 ......y = (-3/10)x + 15/10 - 1  .....y = (-3/10)x + 1/2

Now we can find  the x intersection of these mediians by setting the equations of these lines equal

(9/2) x - 15/2  =  (-3/10)x + 1/2

(9/2  + 3/10) x  =  (1/2  + 15/2)

(24/5)x = 8 

x = 8 (5/24)  =  40/24  =  5/3

And the y coordinate of the  intersection of these medians  is

y = (-3/10)(5/3) + 1/2  =  0

So.....the  intersection of these medians =  (5/3 , 0)=  G

Now....the midpoint of  AB  = [(-5+ 3)/2, (2+6)/2) ]  = (-2/2, 8/2)  = (-1,4)  = F

And ths slope thruogh FC  = [ -8- 4]/ [7-  - 1] =  [ -12/8]  = -3/2

So the equation of the  line  through  FC  .....y = (-3/2) (x - -1) + 4.....y = (-3/2)x - 3/2 + 4....y = (-3/2)x +5/2

So...to prove  that   G  is on this line...let  x = (5/3)

So  when x  =(5/3)  then y  = (-3/2)(5/3) + 5/2  =  -5/2 + 5/2  =   0

So   ( 5/3 , 0)  is on FC  whivh proves that all therr medians intersect at G = (5/3, 0)

Here's a pic :

Let the amount of peanut oil =  x quarts

Then the amount of olive oil  = (1.3  - x)  quarts

Then we can solve this equation

.19x  + .71 ( 1.3 - x)   = .39 ( 1.3)        simplify

.19x + .923 - .71x  = .507

(.19x  - 71x)  =  .507 - .923

-.52x  = -.416     divide both sides  by  -.52

x   =  .8   (quarts  of peanut oil used)

The perimeter  is given by :

2* radius  + arc lengrh of the  sector  =

2r   +  r* theta      where  theta  is the measure  of the central  angle of the sector ( in radians)

So   we  have  that

38  =  2r  + r * theta     simplify

38 = r ( 2 + theta)    subbing   in for the radius  we  have that

38 = 9 (2 + theta)    divide both sides by  9

(38/9)  = 2 + theta     subtract 2 from both sides

20/9 =  theta  

The area of  the sector =   (1/2)* r^2 * theta

So....the area  =    (1/2) (9)^2  * ( 20/9)  =   (1/2) ( 9) (20)  =   90 cm^2

The sum of the first  n  even numbers   =    n ( n + 1)  

We have 250  even numbers  from  2 to 500  inclusive  

So....the sum of these  = 250 * 251   =  62750      (1)

The  sum  of the  even numbers from  2 to 8 inclusive  =  the sum of the first four even numbers =  4 * 5  = 20  (2)

So....the sum of the even numbers  between 10 and 500 inclusive  =  (1)  - (2)  =  

62750 -  20   =

62730

Let  Kaye's  age  now  =  K

Then Cristina's age  =  3K -10

In two years we  have  that

3K  - 10  + 2  =  2K      simplify

3K - 8  =  2K        

3K - 2K  =8

K  = 8     .....Kaye is 8 yrs old

Check

Cristina's age  now  = 3(8)  -10  =  14

In two years Cristina =  16

And this is twice Kaye's age now