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Great job, Chris!

(b)      Let x  be the mL of A

Then  100 - x  is the  mL of B

And we have that

.80x  + .30 (100-x)  =  .50(100)   simplify

.80x   + 30  - .30x  =  50

.80x - .30x  =  50  - 30

.50x  =  20       divide both sides by .50

x = 40  = mL of A

100 - 40  =  60  mL  of B

Amazing, jfan!

OK....here we go  .....

(a)  Let x  be  the   mL of Solution A that we need

So....we  have  this equation 

.80x  +  .30 (500) = .70 ( 500 + x)    simplify

.80x + 150 = 350  + .70x

.80x - 70x =  350 - 150

.10x=  200    divide both  sides  by  ,10

x =  2000  (mL  of  Solution A  )

Good answer, Cal   !!!

I fixed it.

We can write the equation:

(x-4) * (2y-1) = 1 +x*y 

which is what the problem gives us

It then asks us to find xy

Let's expand out the left side first. We get:

2xy-x-8y +4 = 1 + xy

Grouping everything on one side, we get:

xy - x - 8y + 3 = 0

Now, we can make handy of a principle known (on aops' forum at least) as "simon's favorite factoring trick", or "completing the rectangle".

if we add 5 on both sides, we can factor the left hand side, into:

(x - 8 ) * (y - 1) = 5

The problem then becomes a lot simpler, since 5 has two integer factors: 5 and 1

Let's assume x - 8 = 5 and y - 1 = 1

Then we have:

x = 13

y = 2

Clearly,  13>10, so this solution is not correct.

Let's assume the other case:

where,

x - 8 = 1

and

y - 1 = 5

Then we have:

x = 9

y = 6

Clearly, this fits our criterion, giving us the answers of 

9 * 6 = 54

No prob !!!

(a) , (b)   can't answer  these  unless we know  the ml  of Solution B

(c) This is impossible.....the strongest possible  solution would  be  a pure solution of  A =  80%

There's no way we  can get  a solution that is 90% acid

Because his average for the four days was 90, that means his total score was 90 * 4 = 360, since average is defined in this context as:

(total score) / days played, which is

(total score) / 4 in this context

That means to get the total score, just multiply his average by 4, and we get 360 total score.

We already know his scores on the first 3 days, which were 93, 92, and 89 respectively. 

To get his score on the 4th day, we just have to add these up and subtract the sum from the total score, 360

We then get:

360-(93+92+89) = 360 - 274 = a score of 86 on the fourth day

Note  that   ±√500 ≈  ±22.36

So....the  integers  between   -22.36  and + 22.36   that  are  multiples of  7  are  :

-21, -14, -7 , 0 , 7 , 14,  21

Note  that  the inclusion of 0  might seem  puzzling....but note  that  0 / 7  =  0  =  an integer result

So....0 is a multiple of 7

Uh cal, that's not what a life insurance policy is, I think you kind of misunderstand. He's not talking about why a "life" is important, but rather "life insurance".

Maybe  this will help  :  https://www.desmos.com/calculator/46w8zosudi

The  domain for cos (x)  is  [ 0 ,pi ]

The range   of cos (x)  = [ -1, 1 ]

Then  the domain for  the  inverse cosine  is  [ -1,1]

And  the  functional values  (the range)  for the  inverse cosine  is   [ 0 , pi ]

Parentheses  do not  include the  endpoint(s)  of an interval

Thus   (0, 3)   include all the real numbers  between  0 an 3, but  not including  0  or  3

Brackets include the endpoints....so  [0,3 ]  include  all real numbers between 0 and 3  including  0 and 3

Note  that  these  could  be combined

So   [0, 3)  includes  0  but  not 3  on the interval

And  (0, 3]   includes 3 but not 0  on the interval

One more note :  infinity  is  never bracketed

Nevermind, I got it. The answer is 9.

good job young `man

( 900+69+99999 )  /  78888 =

100968 / 78888 =

1.279890478....

Assuming that 78888 is being divided by the sum of 900, 69, and 99999, you would add first, and then divide afterwards.

Awesome, CPhill!!!!

See the following pic :

Let's  find AC =  √ (BC^2  -  BA^2)  = √ (15^2  -12^2)  = √ [225 - 144] = √ 81   =  9

Because  ABC   is bisected, we  have the  folllwing relationship :

AD /DC  =  AB/ BC

AD/DC  =  12/15  = 4/5

So

AC has 9 parts  and  AD  is 4 of them....so   AD  = 4

And  DC   = 5

Using the Pythagorean Theorem

BD  = √  (BA^2  + AD^2)    = √ ( 12^2  + 4^2) =  √ 160   = 4√ 10

Note  that the  sine  of  ADB   = sine of  BDC

sin ADB   = AD / DB  = 12 / (4√ 10)  =  3/√ 10

And note that BDC is  obtuse, so the  cosine will be negative

 And   cos  BDC    = -√ [ 1  - ( sin ADB)^2]   =  √ [ 1 -  (9/10)]  =  -  √ ( 1/10)  =  -√ 10/10   

CORRECTED  !!!!

That means one side length of a triangle is 4/3.

4/3 times 6 (The sides of a hexagon) would be 24/3 which is 8.

Hence, the answer is A.

Hope this helped!

Since she took away $93, -93 would represent it! 

Hope this helped!

774 pi is the answer 100 percent correct!!

just as good, thanks!

x =  cost for lb of  beans

y =cost  for lb of peppers

3x  +  2y =  7.05        multiply through by  3  ⇒  9x + 6y  =  21.15

4x  +   3y =  9.83      multiply through by  -2  ⇒ -8x  - 6y  = -19.66     add  these and we have

x = 1.49    

And

3(1.49)  + 2y = 7.05

4.47 + 2y  = 7.05

2y = 7.05  - 4.47

2y = 2.58  divide  each side by 2

y = 1.29