CPhill

We have  that

(1)^4  + a(1)^3 + b + c =  14

(-1)^4  + a(-1)^3  + b(-1)  + c =  0

(-2)^4  + a(-2)^3  + b(-2) + c = -18      simplify

1 + a + b + c = 14

1  - a  - b+ c  = 0

16 - 8a  -2b + c  = -18

a + b + c =  13      (1)

-a - b   +  c =  -1     (2)

-8a - 2b + c = -34   (3)

Adding (1)  and (2)  we get that

2c= 12

c = 6

Multiply   (1)  by  2   and we have

2a + 2b +2c   = 26

2a + 2b + 2(6)   = 26

2a + 2b  =  14     (4)       

And    (3)  becomes

-8a  -  2b + 6 = -34

-8a - 2b  = -40     (5)

Add (4)  and (5)

-6a  = -26

a = 26/6 =  13/3

And

a + b + c  = 13

(13/3) + b  + 6  =13

31/3  + b = 39/3

b  = 8/3

{a,  b,  c }  =       { 13/3, 8/3 , 6 }   

Radius of cone

16pi = 2pi r   

r = 8

192 pi  =  (1/3) pi  8^2  ( new height)

192  = 64/3  * (new height)

192 * 3 /64   =  new height

(192/64) * 3   = new height

9 in =   new height

Ratio  =   9  / 80  

WYZ  and WXY   form  right triangles

If  YZ  =15

And tan Z   =  WY/ YZ   = 4/3

Then WY=  (4/3) YZ =  (4/3) (15)   =  20  =   height of trapezoid

If  WY  = 20

And tan X = 2 =  WY/ WX

Then     20 / WX = 2    →   WX  = 20/2   =10

Area of trapezoid  =   (1/2) WY  ( YZ + WX  )  =  (1/2) (20) ( 15 + 10)  =  10 (25)  =  250

floor (r)  +  r  = 13.2

No solutions

If      5 <  r <  6

Then the max value for  the  sum  is   almost  12

If   6 < r < 7

Then the  max  value for the  sum  is  almost 13

But if r  =  7.....the sum  = 14

AB= 4AD  = 9BC

AB/ 4   = AD

AB/ 9 = BC

So

AB/4  +   AB/9  =    AD  +   BC

[ 13/36]AB =  AD + BC

So 

CD =  AB- [13/36]AB  =    23/36   AB

So....the probabi;ity is    23/36

(n - 2)  (180) / n  = 144

(n - 2) (180)   = 144n

180n  - 360  = 144n

180n - 144n  =  360

36n  = 360

n = 10

I looked at this the other  day, jenny....unless I blew a fuse (always possible)....I didn't  find any  max

Here's what I  did

h =  300/ (pi r^2)    →  h^2  =  (90000/ pi^2)r^-4

Subbing   x for  r   we have  that

S   =  (pi * x)  ( x^2  +  (90000/pi^2)x^-4 )^((1/2)

The Calculus is messy  but this graph  shows  that  there is  no  positive  x = r   that will  produce a max.....only  a  min

https://www.desmos.com/calculator/zpyqlutuoq

A radius of  ≈  4.07 cm    will produce a minimum surface area  of  ≈ 90.229 cm^2

h =  300 / ( pi * 4.072)  ≈  5.76  cm

P.S.   - check my math....I could have missed something 

*******

Been good.....just took a little "break"  ( for about  5 months....LOL!!!!)

Correct  !!!!

Good to see you, jenny   !!!!