CPhill

This will  evaluate  to    

            1

________________________  =  

x * x^2  * x^3 * .........x^9*x^10

                1

_________________________     = 

        x^ (1 + 2 + 3 +... +9 + 10)

                    1

____________________   = 

          x^[ 10 * 11  / 2 ]

                     1

              ________

                   x^55

So

                 1

        __________

               2^55

We will multiply 1000000  by (5/2) four times (to get the 9th term) and then multiply this result by (1/2) to get the tenth term...so....

1000000 (5/2)^4 * (1/2)  =

500000 * (5/2)^4

500000 * (2.5)^4    =      19,531,250

4331  + 1052  +  ( 459 - N /2)  = 5543

5383  + 459  - N /2  =  5543

5842 - N /2  =  5543

-N / 2  = 5543 - 5842

-N/ 2 =  -299

N/ 2  = 299

N = 299 (2)  =   598

See the graph here

https://www.desmos.com/calculator/nztbhkeg63

The total area  is a square with a side of 1

We have a right triangle  with  legs of  (1/4) and (1) which is  not  included in the inequality region

Its area  =  (1/2)(1/4) (1)  =  1/8

So....the probability  that   y ≤ 4x  is

[ 1 - 1/8)] / 1  =   7 / 8

BuilderBoi  already found the area to be 25/9

So the side of the square =  sqrt (25/9)  = 5/3

Here's the proof

The  top of the  square  has the length   5/9 + 5/9 + 5/9 =  15/9 =  5/3

By symmetry  the   top edge of one of the two rectangles at the bottom = (5/3) (1/2) =  5/6

And the area of one of these rectangles =  5/9

So  the other dimension  of one of these rectangles must be  (5/9) / (5/6) = 6/9 =  2/3

And AB =  1

So....the  side of the  square is also   1 + 2/3   =  5/3

4- 8i

______      multiply numerator/ denominator by   4 - 8i

4 + 8i

(4 -8i)^2

_________

16 - 64i^2

16 - 64i + 64i^2

____________

        80

-48 - 64 i

_______

    80

- 3  - 4i

_____

     5

  (3 + 4i)

________

     -  5

Very nice, BuilderBoi !!!

Add all the equations and we get

10a + 10b + 10c + 10d =  0

So   dividing through by 10

a + b + c + d  =  0

Multiply through by x^2

6x^2  = 35x  - 49  + 5      

6x^2  = 35x - 44      rearrange as

6x^2 - 35x + 44  =  0

(x - 4) ( 6x - 11)  =  0

x - 4 = 0                 6x - 11  =  0

x  = 4                       6x  = 11

                                 x  =11/ 6

x =  11/6  ,  4

Let's look at the times when it is at 4m

-4.9t^2  + 18t  - .4  =  4 

-4.9t^2 + 18t  - .4 - 4  = 0

-4.9t^2 + 18t - 4.4  = 0          multiply through by -10

49t^2  - 180t  + 44  = 0

Using the Quadratic Formula

             180  ± sqrt [ 180^2 - 4(49) (44) ]             180  ± sqrt [ 23776]

t =        ____________________________ =    _________________

                              2 (49)                                                    98

The time when it is above   4m  is given by

sqrt [ 23776] / 98    -  - sqrt [ 23776] / 98  =

2sqrt [ 23776 ] / 98  = 

sqrt [ 23776]  /  49       sec