CPhill

The common  ratio  =  5/a

The  sum  =          4 / ( 1 -5/a)  =  (4)  /  [ ( a - 5) / a ] =     (4a) / (a - 5)

The smallest integer value of  a  that produces a perfect square is when   a =   9

Angle B  =  180 - 53 - 44 =   83°

By the Law of Sines

BC  /sin A = AC /sin B

BC = AC sin A / sin B

BC  =   11* sin (53°) / sin (83°)   ≈    8.85

(-8, 8)  should be on  both lines.......but it cannot be on the  first line  because

-8 + 5(8)  =  32

But  this should =  7

7! + 6! =   5760

1 | 2 | 3 | 4 | 5 | 6 | 8 | 9 | 10 | 12 | 15 | 16 | 18 | 20 | 24 | 30 | 32 | 36 | 40 | 45 | 48 | 60 | 64 | 72 | 80 | 90 | 96 | 120 | 128 | 144 | 160 | 180 | 192 | 240 | 288 | 320 | 360 | 384 | 480 | 576 | 640 | 720 | 960 | 1152 | 1440 | 1920 | 2880 | 5760 (48 divisors)

Count 'em up  !!!!   [easiest to  just count the odd divisors and  subtract this number  from  48   ]

The slant height of each cone =  the radius of the original circle 6 in

The circumference of  each  cone at the base =  Circumference of the  circle /  4 =   (2pi) (6)  / 4 =     3 pi

C = 2pi r

3 pi  =  2pi * r                  divide both sides by 2 pi

3/2  = r =   1.5 in

The height of each cone  = 

sqrt  [ slant height ^2  - radius^2 ] =

sqrt  [ ( 6)^2   -  1.5^2 ] =    sqrt   [ 36 - 2.25 ]  ≈  5.8  in 

See the graph here : https://www.desmos.com/calculator/k4kxivfue1

This forms a  triangle  with a base =  8 - - 6 =  14

And a height of 2 + 1/3  =  7/3

The area is   (1/2) (14) (7/3)   = 49 / 3  units^2

The slope is

[ 2 - 7 ] / [ 3 - -1 ] =  -5 /4

Using the second point and the  slope to  write the equation of the line

y= -(5/4) (x -3) + 2

When x = -6

y = (-5/4) ( -6 - 3) +  2

y = (-5/4)(-9) + 2

y = 45/4  + 8/4

y =  53 / 4

See alan's answer here :  https://web2.0calc.com/questions/please-help_63707

12, 15, 18, 21, 51 , 81, 102 , 105 , 108, 111

The sum of the roots =  -(-11) / 5 =  11 / 5

So

1 / (a + b)  =  1 / (11/5)  =   5 /11