CPhill

Call the number of additional slips showing a '4", x

So we have

2 [  8 / (10 + x) ]   +  4  [ 2 + x ] / [ 10 + x ]   = 3.5         simplify

16 / (10 + x)   +  [ 8 + 4x] / (10 + x)   = 3.5

16  +  8  +  4x    =   3.5 ( 10 + x)

24 + 4x    =  35 + 3.5x

.5x   =   11

x =  22

2 * [ 8 / (10 + x) ]   +  4  [  x + 2] / [10 + x]  =  3

Where x is the number of additional slips  showing "4"

Simplify the above.....and we get

[ 16 +  4x + 8 ] / [ 10 + x ]  = 3

24 + 4x      =  3 [ 10 + x ]

24 + 4x = 30 + 3x

x =  6

2 * (8/10)   + (4 + x) (2/10)  =  3

Where x is the number of additional slips showing "4"

Simplify the above.....and we get

8/5  + (4 + x) (1/5)   =  3

8/5  + 4/5   +  x /5 =  3

12 / 5 + x / 5 =  3      [ multiply through by 5 ]

12 +  x   =   15

And it's obvious that x =   3

OK....!!!

arcsin   [ (1017.5) * sin (145) / 2035 ]

Paste this into this website  and hit "enter"

https://www.wolframalpha.com/input/?i=arcsin%C2%A0+%C2%A0%5B+(1017.5)+*+sin+(145)+%2F+2035+%5D

I'm using   x for a and y for b

Look at the graph, here :  https://www.desmos.com/calculator/5iayh796gi

The  region formed by the first two inequalities is 24 units^2

The region formed by ab > 0 ⇒ xy > 0    is split into two rectangles

One is 3 x 2   = 6 units^2

And the other is 1 x 4 = 4 units^2

So....the probability that   ab > 0   ⇒  xy > 0  is  

[ 6 + 4 ] / 24    =  10 / 24    =    5 / 12

cos (theta)  = -2 / √5

sin theta =  1/√5

tan (theta + pi/4)   =

sin (theta + pi/4)

_______________  =

cos (theta + pi /4)

sin theta * cos pi/4   + cos theta * sin pi/4

_________________________________ =         

costheta * cos pi/4 -  sin theta * sin pi/4

 { sin pi/4 = cos pi/4 =  1 /√2 }

1/√2  [ sin theta + cos theta ]

_______________________  =

1 / √2 [ cos theta - sin theta ]

[ 1/√5  + (-2)/√5 ]

_______________   =

[ (-2)/√5  - 1/√5 ]  

[ 1 - 2 ]                  -1                 1 

________    =      ____     =    ___

[ -2 - 1 ]                  -3                3

See this graph : https://www.desmos.com/calculator/3psnr4vmhw

The area covered by the first two inequalities is   18 units^2

The  inequality    x + y ≤ 4   forms a trapezoid inside this region

Its   area   is   (1/2)(3) ( 4 + 1)    =  15/2   units^2

So....the probability that a point is chosen inside the trapezoid  in the feasible region is

(15/2) / 18   =    15/ 36      =  5 / 12

Let the side of the cube be, S

HA  = S√2

AC = S√2

CH = S√2

So.....HAC forms an equilateral triangle.....and each angle = 60°

So....the sin 60°   = sin HAC =   √3 / 2 

I just got lucky.....LOL!!!!