CPhill

-2/9-(-2)  =

 [ - 2 / 9]  + 2             [2 = 18/9]

 [ - 2 / 9] + [18 / 9]  =

16 / 9

Note :     10 + 1/2  =   [21/2]     and [21/2] * (1/2)   =    21/4   =  5 + 1/4 

So.........the amount  of cedar he normally uses  has been halved

......then.....the normal amount of redwood just needs to be halved, as well

So.....

6 + 1/3  =   19/3      and   [19/3] * (1/2)  =  19/6  =   3 + 1/6  ft of redwood

Let's compute the perimeter  =  2 (L + W)

And we have   2 (9/64 + 1/32)   =      ......  [1/32 = 2/64] ........

2 (9/64 + 2/64)  =

2 ( 11/64)  = 

22/64    [do not reduce !!!]

Length / Perimeter  =    [9/64] / [ 22/64]  =  [9/64] * [64/ 22]  = [64/64] * [9 / 22]   =  1 * 9/22 = 9 / 22

Width / Perimeter  =    [2/64] / [ 22/64]  =  [2/64] * [64/ 22]  = [64/64] * [2 / 22]   = 1 *  1 / 11  = 1 / 11

Awesome!

Could you just explain to me how you go from:

sqrt (k + x ) = 7  - sqrt(x)   square both sides

to

k + x  =     49   -  14 sqrt(x) +  x  

+++++++++++++++++++++++++++++++++++++++++++++++++++++++

OK.....squaring both sides, we have

(√( k + x) )^2  = ( 7  - √x)^2      [squaring the left side just causes the radical to disappear]

k + x  =  ( 7  - √x) ( 7  - √x)    expand on the right

k + x  =  49  - 7√x  - 7√x  + x

k + x  =  49  - 14√x    + x   

Does that help ??

OK...I see....let me get you there

sqrt (k + x ) + sqrt(x) - 7 = 0     rearrange

sqrt (k + x ) = 7  - sqrt(x)   square both sides

k + x  =     49   -  14 sqrt(x) +  x        subtract x from both sides

k = 49  -  14 sqrt(x)   rearrange

14 sqrt(x)  =  49  - k  square both sides

196x  =  2401  - 98k + k^2     divide both sides by 196

x  = [2401  - 98k + k^2] / 96  =  [ k^2 - 98k + 2401] / 96 

Voila....!!!!!

infinity  + 1   =  infinity

sqrt (k + x ) + sqrt(x) - 7 = 0

Note, Tony.......there are  infinite things that might work, here.....because we have more variables than equations......for example...

k = 49    and x  = 0

sqrt (49 + 0 ) + sqrt(0) - 7 = 0

sqrt (49 ) + 0  - 7 = 0

7 + 0 - 7   = 0

or

k = 7  and x = 9

sqrt (7 + 9 ) + sqrt(9) - 7 = 0

sqrt (16) + sqrt(9) - 7 = 0

4   +  3    - 7 =  0

7  - 7 = 0

etc  .......

This willl always happen whenever we have more variables than equations.......

(5 - sqrt ( -59) ) / 6

(5+ sqrt (-59) ) / 6

The concept of "positive" and/or "negative" − in real number terms −  has no meaning here, because both of these will be complex solutions......

(k-2) x^2 -5x +7 = 0

Knowing that P = 7  (P = c/a)

c /a    =   7 / (k − 2)  =  7 / 1

This implies that .....

k − 2   = 1

k = 3

Call the amount that the first cask can hold  = A  .... and call the amount that the second cask can hold  = 2A

So the amount of 3 year old wine in the first cask = A/2....   and the amount of 3 year old wine in the second cask  = (1/3)2A  = (2/3)A

So....if he contents are mixed....the part that is 3 year old wine  =

[ A/2   +  (2/3) A ]  /  [ A + 2A]   = 

[ (3/6)A  +  (4/6)A ] / 3A =

[ (7/6)A]  / 3A  =

[ (7/6)] / 3  =

7 / 18