CPhill

x^2  +  18x + 6x  +  ____

x^2 + 24x  +   _____

Take  (1/2) of 24   =12

Square it  = 144

That's the missing  number

Proof

(x + 12)^2  =  x^2 + 24x + 144

P.S.  - this proceedure is  known as "completing the square "

x^2  +    __x + 100 + 44

x^2 +  ___x   + 144

Take  both square roots of  144   = 12   and  -12

Multiply both by 2  =   24   and  -24

Those are the  x^s  we need

Proof

(x - 12)^2  = x^2  - 24x  + 144

(x + 12)^2  = x^2 + 24x + 144

Assuming  that   letters can  be  repeated....we  can  choose   any of the 26 letters for the  first  three places  and  any of the remaining 25  for the last one

So

26^3 * 25   =  439400

LOL!!!

Never forget   the  "pie"   !!!!!

Oops...I put in the incorrect volume....see my corrected answer  !!!!

Here's my best attempt

Since A  joins C   (without overlap)   the circumference of the  cylinder  =  6

So   the radius  can  be figured     as   

6  =  2pi * r

r  = 3  / pi

And since we know the volume of the  cylinder  we  can  figure the height as

(27/2) =  pi  (3/ pi)^2  * h

(27/2)  =  9 / pi  * h

h =  ( 27pi)  / (2 * 9)  =    27pi /18  =   3 pi /2

So.....this also   =  the  rhombus height

And the sin BAD  =   rhombus height  / rhombus side   =   (3 pi/2) / 6  =   pi  / 4

Corrected   !!!!

LOL!!!....not always   !!!!!

If both have the same  number at the end  then  each  has   (772/2)   = 386

Let  E  be the number that Eugene started with and F what Frank started with

And   F  =  772  - E        (1)

Eugene gives away  (1/3) of his

And from Frank he gets   (1/5) ( F  + (1/3) E ]  =   (1/5) ( 772  -E + 1/3)E )     (2)     

So  at the end,   what  Eugene started with  - what he gave away + what he got from Frank  = 336

So

E  - (1/3)E  +  (1/5) (772 - E + (1/3)E  ]  = 336   

(2/3)E +  772/5 + (1/5)(-2/3)E  = 336

(2/3)E - (2/15)E  =   336   - 772/5

[ 8/15] E  =  908 / 5

8E  =  2724

E = 340.5   ?????

Are your numbers correct  ???  (it's also possible that I made an error !!! )

Because of the   leading x^4 term,  the max  is unbounded    =    inf

Using some Calculus  to find the min

q'(x)  = 4x^3 + 8x - 24x

q'(x)  =  4x^3 - 16x

Set this  =  0

4x^3 -16x   = 0

4x (x^2 - 4)  = 0 

x=  0     or  x   =2

Taking the second derivative

12x - 16

When x = 0     this is negative .....so....not a min here

When x =  2  this is positive, so  x is a min here

Petting 2 back into  the  function and evaluating  gives us    -30   

So

Range  =   [ -30 , inf )