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Not sure about the  first one  .....but....for the second one

We have a  same degree polynomial . same degree polynomial   situation....so.....the  limit  of  nth term as x  becomes increasingly large  is just the ratio  of the coefficients on the highest power terms in the numerator and  denominator....so  we  have

8 / 6    =  

4 / 3

really i thought it was 983989824982984984

Equation of first circle  :

(x - 2)^2   + ( y + 1)^2  = 16

Equation of second circle :

(x - 2)^2   + (y - 5)^2  = 10

Subtract  the second equation from the first and we have

(y + 1)^2  - ( y - 5)^2  =  6           simplify

y^2 + 2y + 1  -  y^2  + 10y - 25  =  6

12y - 24  =  6

12y =  30

y = 30/12 =  2.5 

Using the first  equation to find the x intersection points, we have  that

(x - 2)^2  + ( 2.5 + 1)^2 = 16

(x - 2)^2  + (3.5)^2   =16

(x - 2)^2   +  12.25  =16

(x -2)^2  =  3.75

(x - 2)^2  = 15/4       take both roots

 x - 2 =  ±√15/ 2

x =  ±√15/2  + 4/2

x  =  [ 4 +√15] / 2         and    x =  [ 4 - √15] / 2

The   distance between  these points  =   [ 4 + √15 ] / 2  - [ 4 - √15] / 2   =  2√15/ 2  =  √15

So.....the square of this distance =  15

x mod 17 = 1
x mod 7   = 2
x mod 31 = 3
x mod 23 = 4
We can use the Chinese Remainder Theorem + Modular Multiplicative Inverse to solve the congruences:

1 - The LCM [ 17, 7, 31, 23] = 84,847

2 - 84,847/17 =4,991, 84,847/7 =12,121, 84,847/31 =2,737, 84,847/23 =3,689 [The quotients]

3 - We have to find the Modular Multiplicative Inverses(MMIs) of the quotients in 2 above:

4 - MMI of 4,991 mod 17 =12, MMI of 12,121 mod 7 = 2, MMI of 2,737 mod 31 = 7 and MMI of 3,689 mod 23 = 18

5 - We multiply the remainders x The quotients in 2 above x MMIs in 4 above and sum them all up.

6 - x =[1.4991.12] + [2.12121.2] + [3.2737.7] + [4.3689.18] = 431,461

7 - We take 431,461 mod(LCM) 84,847 =7,226 - which is the smallest positive integer that satifies the 4 congruences.

8 - The full answer is: LCM [84,847n + 7,226], where n =0, 1, 2, 3.......etc.

The possible rational zeroes  will  be

 ± (  All  factors of  8) / (All factors of 2)

Factors of 8  =

1, 2, 4, 8

Factors  of 2  =

1, 2

So....the possible rational  zeroes  are

 ±  ( 1/1, 2/1, 4/1, 8/1, 1/2)  =

 ± ( 1/2 , 1 , 2,  4 , 8)   

Well.....I don't know what is assumed here.....but....

If  the race is  only  over the length of the pool ......then  Ario  will  swim  5m   before  Miguel starts

To  see this    he swms   1(5)  =  5m    and he has  20 m  left to swim

So    5 : 20  =   1 : 4

If the race  starts  3 m  from the pool then both will  travel  25 + 3   =  28 m

So.....there are 5  equal segments to the race....each being  28/5 =  5.6m

So.....Ario will  travel   (1 * 5.6)  = 5.6  m  before Miguel starts

And he has 4 (5.6) =  22.4  meters left to travel

Note that  5.6 : 22.4  =  1 :  4

But....the distance that he actually  swims  over the  5.6 m  =  5.6  - 3  =  2.6 m    

Second one

16  +  32/3  +  64/9  +  128/27 +  .......

This is a convergent  infinite sum

The first term is   16

And the commom ratio  is        (32/3) / 16  =   32/48  =  2/3

The  sum  is

      16                 16

________  =      ___  =      48

   1  - 2/3             1/3

First one.....by the ratio test we have

lim                 4 ^{(n + 1) + 1}                      4n + 1

n  →  ∞      ____________  *  _______   =

                    4 (n + 1) + 1            4^{n + 1}

                        4 ^{n + 2}                 4n + 1

lim                 ________ *     ________    =

n  →  ∞           4 ^{n + 1}                  4n + 5

lim                     4   *   1   =        4    

n  →  ∞

Because this limit   > 1, the series diverges

cry some more

< which is the more accurate way to estimate 48% of 23 >     

I'm taking you at your word you mean estimate, rather than calculate. 

Here's what I do.  I see that .48 is the same as .50 – .02  

In my head I take .50 of 23  ......       11.50 

Simultaneously take .02 of 23  .......     .46     (i.e., 2 x 23 is 46 then move the decimal point two to the left)

This I just subtract 11.50 – .46 and get 11.04   

This one was simple and happened to come out exact.  

The same approach works with other values with possibly less accuracy. 

_.

1 - Tevin reaches his maximum speed 5 seconds and 35 meters from the start. But 67 meters from the start and 9 seconds later he continues the race at his maximum speed.

He attains his maximum speed at the start of the 36th meter into the race and after 5 seconds from the start. Therefore, his maximum speed was: 67 - 35 =32  divided by 9 seconds - 5 seconds for the first 35 meters. Or:

32 / 4 = 8 meters per second was his maximum speed. He continued at that speed till the end of the race, or the distance of 100 meters.

2 - If he continued at that rate for another 3 seconds, he would have covered a distance of: 8 x 3 =24 meters

See my answer here  :

https://web2.0calc.com/questions/urgently-need-help-within-two-days

GingerAle may be a mean, trolling b i t c h, but she is very good at mathematics and she excels at statistics. I’d wager a hundred dollars in BitCoin that her answer is the correct one.  It’s clear, concise, and looks like a textbook answer. Anyone with half a brain can do similar questions by using her answer as an example.  

GingerAle also signs her name to her well written, hilarious, mean b i t c h troll posts, instead of hiding anonymously as a guest, like you pansy asses!!

a.

Speed = Distance/time

He ran 67 meters from the starting line in 9 seconds. 

\(speed=\frac{67}{9}=7.444m/s\)

b.

Well if he runs 7.4444...m per one second 

If he ran for an additional 3 seconds with that max speed then 

67/9*3=22.33333333 meters in those 3 seconds

Thus, the total distance he travelled is 67+22.333333=89.33333333333m

a stop sign is a regular octagon formed by cutting triangles off the corners of a square. if the stop sign measures 36 in from top to bottom.   What is the length of each side of the octagon.

There are triangles with side lengths 1, 1, sqrt(2) and 2, 2, 2*sqrt(2).  So there are a total of 12 + 4 = 16 triangles.

In order for the area of triangle ABD to be at most 12, the height of triangle ABD is at most 2.  The height of triangle ABD can be 6, so the probability is 2/6 = 1/3.

Thank you so much!!

Thanks so much! I will try to figure out any simpler ways, great job on your answer! I suspect that guest is the one who is falsely posting answers, as melody as stated on one of my other posts.

If these answers are right then that means the mean trolling bietch Gingerale’s answer is wrong !!

https://web2.0calc.com/questions/help_75920 

My result is as follows:

Though there must be a simpler way!

Thank you, Dragan !

Thank you, CPhill !