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The y axis intercept occurs when x = 0    so put x = 0 into the equation and solve for y value

y^2 -10y + 25 = 4y + 16

y^2 -14y +9 = 0

   If you use the quadratic formula to find y = two roots   so the equation of the circle intersects the y -axis at two points....

  y=  7 +- 2 sqrt (10)      and x = 0 for each of the points

What  all  4 took  must be all the berries in the  bowl....call this N

Billy took (1/2)N + 2

Bob  took  (1/3)  [ (1/2)N  + 2 ] + 2

Joe took   (1/3)   [  (1/3)  [ (1/2)N  + 2 ] + 2 ]  -1

Mom  took 1

So we  have  this rather convoluted equation

[ (1/2) N + 2  ]+    (1/3)  [ (1/2)N  + 2 ] + 2  +  (1/3)   [  (1/3)  [ (1/2)N  + 2 ] + 2 ]  -1 + 1  =  N

Because  this is a little tedious  ( but not impossible) to solve, I'll get a little  help from WolframAlpha

N  = 20

Billy took  (1/2) (20) + 2  = 10 + 2  =   12

Bob took  (1/3) of what Billy took + 2 = (1/3)12+ 2  =  4 + 2  =  6

Joe took   (1/3)  of what Bob took and put one  back  =(1/3)(6)  - 1   =   2  - 1  =  1

And Mom took 1

And   12  + 6 + 1 + 1    = 20 berries in the  bowl originally

x = original number of berries

x -   1/2x  -2    -  1/6  - 2    - 1/18 x   = 1

x - 13/18 x   - 4 = 1

5/18 x = 5

x = 5 * 18/5 = 18 berries originally

    See Chris answer below.....  I am not sure I interpreted the Q correctly

                                                      I interpreted  'he took 1/3 of what Billy took and 2 extra berries.'  to exclude the 'extras'  etc

It  will pass through 13 lattice points

To  see this, it will  pass through a lattice point   at  every x value from  0  to  12

And  these will  be paired with y values  36, 33, 30 , 27.......0

So....13 lattice points

4x + 6y + 13 + 2x   -2y   simplifies to

6x + 4y + 13

So

x^2 + y^2  ≤  6x + 4y  + 13

x^2  -6x  + y^2 -4y  ≤ 13       complete the square on x and  y

x^2  - 6x + 9  +  y^2  - 4y + 4 ≤ 13 + 9 + 4

(x - 3)^2  + (y - 2)^2  ≤  26

This is just the  area  inside  the circle  with a center of  ( 3, 2)   and a radius of  sqrt (26)

See here  : https://www.desmos.com/calculator/0c8kf8nfll

The area  is    pi (sqrt (26))^2 =    26 pi

True....equal...

Pure  acid   =100%  acid  =  1

So...let  x  be the amout of pure acid to  be added and we  have that

1x +.10 *( 90 - x)  =  ( .58)(90)

x  + 9  - .10x  =  52.2

.90x  =  52.2  - 9

.90x  =  43.2      divide both sides by  .90

x  = 43.2  / .90  =  48   L

thanks a lot bro, and for the last one, it is equal right? i just wanted to make sure

thanks a lot bro, and for the last one, it is equal right? i just wanted to make sure

That is pretty cool   !!!!

25/5  =   5 zeroes

25/5^2 =  25/25  = 1  more zero

So

5 +  1    =  6 zeroes

No odd ones are  divisible  by  8

So   we  have

8^2  = 64   = 8 * 8

12^2  = 144  = 8 * 18

16^2   = 256  = 8 * 32

20^2  = 400  =  8 * 50

24^2  = 576  =  8 * 72

28^2    =  784  =  8 * 96

b = 50 (2)^h

 If the bacteria continue to grow at the same rate, there will be 800 bacteria after 4 hours. 

Need to know how many  cups  the full pitcher holds, Alston, to give a definitive answer

However....let's suppose that  the full pitcher holds  N cups

Ater 1/8N  is  poured out  .....there remains   (7/8)N  cups

The  amt poured out to water the plant

So  (1/4) (7/8)N   =  7/32N    

So....the amt  that remains   =   (7/8  - 7/32)N  =  ( 28/32  - 7/32)N  =  (21/32)N

2.   Arithmetic  series

Common  difference, d,  = 4

First term   =  -3

Explcit rule

a_n  =  -3  + 4 ( n -1)     where    a_n is the  nth term

3. 100th term  for number 2  is      -3 + 4(100 - 1)   =   -3   + 4(99)   = 393

4.  Geometric

common ratio, r  = -2

First term   = -3

Explicit rule

a_n  = -3(2)^(n -1)

5. Geometric....commom ratio   =  -3    first term  = 5

Explicit rule

a_n  = 5 (-3)^(n -1) 

6.    4* 4^(n -1)   =  4 * 4^n * 4^(-1)   =  4 * 4^n   /  4  =   4^n

$\frac{1}{h^6}$

   24.07

+ 16.58

_______

   40.65

It's a repeat question from about a week ago.

Thanks Alan,

That is super cool !

I do not ever remember seeing a problem like that one before.  I like it!

Sorry, I shoud have provided more verbal explanation.

1st step: Invert the fraction and set 'a' to be the largest integer less than or equal to the result.

2nd step.  Subtract 'a' from the inverted fraction.  If this isn't zero

3rd step:   Repeat the first two steps with the fraction left over from step 2, to get the next constant until step 2 results in zero. 

Thanks Alan, but I have no idea what you just did.   

Like so

of degree what?  Is there supposed to be a number there?

ok Thanks Alan,

Sorry Country03, I see your original question did show this clearly, so it was my error.

but when I asked if  A or B was correct you said B, which was not what you wanted at all.

This is what you wanted.

[ (tanx)^2 -1 ]  / (secx)^2  = (tanx - cotx ) / (tanx + cotx)