CPhill

Rearrange as

4x + 4y - xy + 115  =  0

4y - xy  =  -4x - 115     multiply through by -1

xy - 4y  = 4x + 115

y [ x - 4]  =  [ 115 + 4x ] 

y  =  [ 115 + 4x  ] / [ x - 4]

Note that when  x = 5,   y =  135

x + y =   5 + 135  =   140

r = (1/4) / 16  =  1 / 64

30th term  = 

(1/4)r^6

(1/4) (1/ 64)^6   =

(1/2)^2 * ( 1/ 2^6)^6  =

(1/2^2)  * ( 1/2^36)  =

1/ 2^38

1 / 274877906944

Here's a very similar problem

https://web2.0calc.com/questions/help-pls_105

14 + 19 > x             14 + x >  19

33 >  x                       x >  19 - 14

x = 32                        x >  5   →  x = 6

32 - 6 =   26

x = 31^2  =  961

V_cylinder  =   pi * r^2 * h =   pi (8)^2  * 14  ≈    2814.9 in^3

x^3 + y^3  = 

(x + y) ( x^2 + y^2  - xy) =  (8)  (x^2 + y^2  + 200)

This implies that  xy  = -200  →  y = -200/x

So

x + y  = 8

x  - 200/x - 8 = 0

x^2 - 8x - 200  = 0

x^2 -8x  =  200

x^2 - 8x + 16 = 200 + 16

(x - 4)^2  = sqrt 216       take both roots

x  - 4  = sqrt (216)

x = sqrt (216)  + 4          or      x = -sqrt 216  + 4

x =  4 + 6sqrt(6)              or      x =    4  -6sqrt (6)

y = 4 - 6sqrt (6)                       y  = 4 + 6sqrt (6) 

One solution is (x,y)  = ( 4 + 6sqrt(6) , 4 -6sqrt (6) )         

If this  is a  circle  A  must =  2

We can  write

2 ( x + 5)^2   + 2(y  - 2)^2  = 50 + 50 + 8

2 ( x^2 + 10x + 25)  + 2(y^2 - 4y +4)  = 50 + 50 + 8

2x^2 + 20x + 50 + 2y^2 - 8y + 8 =  108

A =2

B = 20

C = -8

r = sqrt (108)

A + B + C + r   =     14 + sqrt (108)

12x^2 + kx + 27 = 8x^2 + 18 + 14x^2 + 32

Rearrange as

10x^2  - kx  + 13  =  0

This will have non-real roots when the discriminant < 0

k^2  - 4(10)(13)  <  0

k^2  <   520

k < sqrt (520)      or   k > -sqrt (520)

( -sqrt 520 , sqrt 520) 

(1)

258  =  129 * r^2

r^2  =  268 / 129

r^2  = 2

r  = sqrt 2

N = 129 * (sqrt 2)^t        where t is the number of years after 2011

(2)

C =  160 * (1/2)^t        where t  =  day t