CPhill

C (8,4)  =  70 ways

A vertical asypmtote occurs  when an x value makes the denominator =  0

So we have  that

x^2 + ax + b = 0                      

And we know that the roots of this  = 1 , 2

By Vieta

The sum of the roots =  -a                 

So                                                                        

1 + 2 =  -a 

3 = -a                                                                                                                                              

a = -3

And the product of the roots  =  b

So

1*2  =  b

2 = b

a + b  =  -3 + 2    =   -1

If y =  0.......then the x intercepts are   3.5  and   -k

If  x = 0   the y intercept  =  .5 (- 3.5) (k)  = -1.75k

So  any  triangle  formed  will  have  the area of 1.5 expressed as

(1/2) ( 3.5 - k) (-1.75k)   =  1.5           multiply through by  2

(3.5 - k) (-1.75 k) = 3      simplify

(7/2 - k) ( -7/4k)  =  3

(7/4)k^2 - (49/8)k  - 3  =  0                 multiply through by   8

14k^2 - 49k - 24  =  0

Using the Quadratic formula

                 49 ± √ [ 49^2  - 4 (14)(-24) ]       

k =         ________________________        ≈   3.96      or   ≈  -.436 

                             2 * 14

We have the form  y = (x - h)^2  + k

(h , k)  is the vertex =    (4 , -10)

This is a parabola that  opens  upward   and the axis of symmetry  is

x =  h   

So

x =  4   is the axis of symmetry

Note that triangles  BYX and   BCA  are similar by AA congruency

And AC = BC  so    XY = BY 

And since BX = 12......then   BX =  sqrt ( XY^2 + BY^2)   =  sqrt (2*BY^2)  =   BY * sqrt 2

So  BY =  12/sqrt 2 =  6sqrt 2

Likewise....since  AC = BC  then

BA = sqrt ( AC^2 + BC^2)  =    sqrt [ 2 * AC^2]  =  AC * sqrt 2 =   18sqrt 2

So  YA =   BA  -BY =  18sqrt 2 - 6sqrt 2   = 12 sqrt 2

So  

BY   /  YA =   [ 6sqrt 2 ] /   [12 sqrt 2 ] =   6/12  =  1/2  

(5/6)D + 9 = D

9 =  D - (5/6)D

9 = (1/6)D         multiply both sides by 6

54   =  total distance

He has traveled   54 -  9  =    45  miles

20  =  (2/3) T   where  T is the total

20 / (2/3)  =  T  =    30  total

Hannah's math teacher put a cup filled with jellybeans on her desk. She then asked everyone to guess how many beans were in the jar. Amanda guessed 193 and was off by 15. Wendy guessed 174 and was off by 4. Emma guessed 198 and was off by 20. Connor guessed 181 and was off by 3. Jason guessed 172 and was off by 6. Jenna guessed 168 and was off by 10. Can you figure out how many beans are really in the jar?

178

Proof

193 - 15 =   178

174 + 4 =    178

198 - 20  =  178

181 - 3 =     178

172 + 6  =   178

168 + 10  = 178

DBC is a right triangle.....so....

BD =   sqrt  [ DC^2 + BC^2 ] =  sqrt [  8^2 + 3^2 ]    =  sqrt (73)

Let  EB = x    and DE = sqrt (73)  - x

And  EBC  and   EDC  are also right triangles

So

CE^2   =     BC^2 - EB^2  =   3^2  - x^2  =   9 - x^2         (1)

and

CE^2  =  DC^2  - DE^2  =   8^2  - (sqrt (73)  - x )^2  =   64 - x^2 + 2xsqrt (73) - 73  = 

-x^2 + 2xsqrt (73) - 9         (2)

Set  (1)  = (2)

9 - x^2  =  -x^2 + 2 sqrt (73) x - 9      simplify

18 =  2  sqrt (73) x 

9 =  sqrt (73) x

x =  9  / sqrt (73)  =  EB

So

CE =  sqrt   [  BC^2 - EB^2  ]  =  sqrt [ 3^2  - (9/sqrt (73))^2 ]   =  sqrt  [ 9 - (81 / 73 ) ]  ≈  2.8

Probability  =  

C (10,6) * (1/2)^10  =  105  / 512   ≈  20.5%