CPhill

A = (4,8)   B  =  (2, -6)   C  = (-4, 4)

We can use the midpoint theorem to find the medians

median of  AB   =   (  [4 + 2]/ 2, [8 + -6] / 2 )  = ( 3, 1)

median of CA  =  ( [-4 + 4 ] / 2 , [ 8 + 4 ] / 2 ) =  (0, 6)

median of BC  =  ( [ -4 + 2 ] / 2 , [-6 + 4 ] / 2 )  = (-1, -1 )

Slope of line  from A  to median of  BC  =  [ 8 - -1] / [ 4 - - 1 ]  =  9/5   

Equation of the median  :  

y = (9/5)(x -4) + 8

y = (9/5)x - 36/5 + 40/5

y = (9/5)x + 4/5        (1)

Slope of line  from B  to median of CA  =  [ -6 - 6 ] / [ 2 - 0]  = -12/ 2  = -6

Equation of the median :

y = -6x + 6       (2)

Slope of  line from  C  to median of AB  = [4 - 1 ] / [ -4 -3]  = [3 / -7]  = [ -3/7 ]

Equation of  the median

y = (-3/7)(x - 3) + 1

y = (-3/7)x + 9/7 + 7/7

y = (-3/7)x + 16/7     (3)

Find the  x intersection of  (1)  and (2) by stting these equal

(9/5)x + 4/5 =  -6x + 6

(9/5)x + 6x  = 6 - 4/5

(9/5)x + (30/5)x  = 30/5 - 4/5

39/5x  = 26/5

39x  = 26

x = 26/39  =  2/3

And plugging  this x value into (2)  to find y, we have  -6(2/3) + 6  = -4 + 6  = 2

So...the intersection point should be  ( 2/3 , 2)

Check this in (1)  and (3)

(9/5)(2/3) + 4/5  =   18/15 + 12/15  =  30/15  = 2

(-3/7)(2/3) + 16/7  =  -6/21 + 48/21  =  42/21  = 2

So.....the medians all pass through  ( 2/3 , 2)

Note that the points  (5, - 2)  and  (4, -3)  would be on the graph  if we had a solid line

So....the slope would  be    [ -3 - -2 ] [ 4 -5]  = -1 /  -1     =  1

So...using   the point (5 , -2)  we can write the equation of this  "line"

We have

y  = 1 ( x -5)  - 2

y = x - 5  - 2

y = x - 7      or     x - 7   = y     subtract  y from both sides

x - y  - 7   = 0

However  ...we have a dashed line so we must have  a  >   or  <  insted of an  =

Note that the point  (0, 0)  is in the shaded region...so....testing this point in

x - y  - 7 >  0

0  - 0  - 7  >  0 

-7  > 0      not true

So...the equation must be

x - y  - 7  <  0

Here's the graph :  https://www.desmos.com/calculator/9jof1vd4lq

Average  rate of change  (per year) =

[ 46.6  - 45.1 ]  / [ 1920  - 1970 ] =

1.5 / -50  =

-.03   crimes  per year

Put everything in terms of powers on 2

A  =  2^(1/2)  / 4^(1/6)  =  2^(1/2)  / (2^2)^(1/6)  = 2^(1/2) / 2^(2/6)  = 2^(3/6) / 2^(2/6)  =

2^(1/6)  = 2^(2/12)

B   =   (128)^(1/12)  =   (2^7)^(1/12)  = 2^(7/12)

C =  [ 8^(-1/5)] ^2  =  [ (2^3)^(-1/5) ]^2  = [ 2^(-3/5) ]^2  = 2^(-6/5)

D  =  √  [  (2)^1 * 8^(1) /   4^(1) ]  =  √  [16 / 4 ]  = √ 4  =  2  = 2^(1)

E   =  [ 2^(1/2)  *  (2^2)^(-1/4) ] ^(1/3)  = [ 2^(1/2) * 2^(-2/4) ]^(1/3)  =

[ 2^(1/2) * 2^(-1/2) ]^(1/3)  = [ 2^0]^(1/3)  = 2^0

So..in order, we have

2^(-6/5) < 2^(0) < 2^(2/12) < 2^(7/12) < 2^(1)

So

C   E   A   B   D

-2 ( s - 7)  >  4s  + 8

-2s + 14  > 4s + 8       subtract 4s , 14 from both sides

-6s > -6      divide both sides by  -6   and reverse the inequality sign

s <  1   

( -infinity , 1 )      [ in interval notation ]

L1   =  5x + 8y  = -9

In the form  Ax + By  = C....the slope of this line  is  -A / B

So...the slope of this line  =  -5/8

And the slope of a line perpendicular to this   =  8/5   = slope of L2

So....we have this equation of L2  that passes through (10, 10)

y  = (8/5)(x -10) + 10

y  = (8/5)x - 16 + 10

y  = (8/5)x  - 6

m  = 8/5     b  = -6      and their sum is   8/5 + (-6)  =  8/5 + (-30/5)  = -22/5

Let S  be the  the total sales per month

So...we have these equations

B  =  900 +  .15( S - 6000)

A = 500 + .10 ( S)

And we want to know when  B > A  ...so...

900 + .15(S - 6000)  >  500 + .10(S)      simplify

900 + .15S  - 900  > 500 + .10S

.15S  > 500 + .10S        subtract  .10S  from both sides

.05S  > 500      divide both sides by .05

S > 10000  ⇒  "B"  is better whenever his monthly sales are greater than $10000

Quadratic  trinomial  is correct

Quadratic = 2nd degree polynomial

Trinomial =  3 terms     [ tri  = 3 ]

Quartic trinomial  [ 4th degree polynomial  with three terms]

Ex :  x^4  + 3x^2 + 1

Cubic trinomial  [ 3rd degree polynomial with three terms ]

Ex :  x^3  + 2x  - 8

Cubic binomial  [ 3rd degree polynomial with two terms ]  [ bi  = 2]

Ex :   x^3  - 8

Let's assume that 220.000  is supposed to be  220,000

Let  salesperson C  earn  E  during this period

Then  B  has earned  50% more than C  =  1.5C  = 1.5E

And A has earned  twice B  =   2B  = 2(1.5)C  =  3C  = 3E

So

A + B + C  = 220000

3E + 1.5E  + E   =  220000

5.5E  = 220000     [ divide both sides by 5.5 ] 

E  = 40000  =  amt  C  earned in the period

And B  earned   1.5C  = 1.5 (40000)    = 60000

So...in any one year, B's average earnings  = 60000 / 3  = 20000