CPhill

Yep......blocked for me as well.......odd  !!!

Nice, hectictar.....!!!

If   3 -  ⁴√2 i    is a root, then so is the conjugate   3 +  ⁴√2 i

Let  the polynomial we are looking for be P(x)

And  suppose that P(x)  can be written as  Q(x) * R(x)

So ....let Q(x) be  formed by

( x -  ( 3 - ⁴√2 i) )  ( x -  (3  + ⁴√2 i) )   =

[ ( (x - 3) +  ⁴√2 i  ]    [  (x - 3)  -  ⁴√2 i  ]   =

(x - 3)²   +  √2

And it's clear that we will have a polynomial with rational coefficients if  R(x)  is

(x - 3)²   - √2

So

      Q(x)            *         R(x)        

[( x - 3)²  + √2 ] *  [( x - 3)²  - √2 ]    produces

(x - 3)⁴   -  2    =

x^4  -  4* x³ * 3  + 6x² * 3²  -  4x * 3³  + 3⁴  -  2   =

x^4  -  12x^3  + 54x^2  - 108x  +  79   =  P(x)

Let C1  be  the constant in Eric's polynomial  and C2  be the constant in Charles's polynomial

And we have that

[16 - 4C1] / [16 - 4C2]  =  C2 /C1     cross-multiply

16C1   -  4C1^2   =  16C2  - 4C2^2

4C1 ( 4 - C1) =  4C2 ( 4 - C2)       divide by 4

C1 (4 - C1) =  C2 (4 - C2)      rearrange   as

C2 / C1  =  (4 - C1) / (4 - C2)  implies that

C2 =  4 - C1       and    C1  = 4 - C2      which  both imply that

C1 + C2  =   4

Rearrange as

x^2 -6y +y^2 - 14y   =  -33      complete the square on x and y

x^2  - 6y + 9    +  y^2 - 14y  + 49   =    -33 + 9 + 49

(x - 3)^2  + (y  - 7)^2  =    25

This is a circle centered at  (  3, 7)  with a radius of 5

The part lying below y  = 7 is the  area of 1/2 of this circle  =

pi * 5^2  / 2   =

(25/2)pi ≈  39.3  units^2

The "formula"  for finding the new points is as follows

New Point  = [  Old Point - Dilation Center ] *Scale Factor + Dilation Center

Here's the first one.....you should be able to do the other three based on this....if not, let me know

Note that  (a, b)  - (c, d)  = ( a - c , b - d)       and  (a , b) + (c , d)  = (a + c , b + d)

Here's the dilation result for  (4,7) 

New Point =  [  (4, 7)  - ( 1, -2) ] * 13  +  (1,-2)

=   [  ( 3, 9) ] * 13  +  (1,-2)

=    (39, 117 )  +  (1, -2)

=  (40, 115)

We can find this

16  +   16 [ 1 - .5^(N - 1) ]  / [  1  - .5]     

Where N is the total distance traveled on the Nth bounce 

When N  = 6, we have

47 ft

Sum of a geometric series  =

a1  /  [ 1  - r ]          where  a1  is the first term   and r is the common ratio....so we have

16  =  a1  /  [ 1  -  -1/5]

16  =  a1  /  [  6/5]       multiply both sides by 6/5

(6/5)16  =  a1   =   

96/5

We  are looking for this

.01 [  1  -  2^N[ / [ 1  - 2]   =  5  

1 - 2^N   =  -500

-2^N  =  -501

2^N  =  501    take the log of each side

log 2^N  =  log 501  

N *  log 2  =  log 501

N  =  log501/log2

N = 8.9   ⇒    9 days ⇒  [ Monday ]

9/36    =   1/4