CPhill

Let x be the days to complete the job in the first case

The amount of work to be done  is  number of workers * numer of days =   5x

So....in the second case

5x  = 6 ( x -15)

5x = 6x - 90

90 = x   =    the number of  days to complete the job in the frst scenario

So

5 * 90  =  n (90 - 40)  where n is the toal number of workers needed in the last case

450  = 50n

450 / 50 = n =  9

So....  9 - 5  =  4 additional workers should be hired

3600 / [ 50 *20 ] =  3.6  sq ft / min  painted by one worker

So

4000 / [ 25 * 3.6 ]  ≈  44.44  min

\( \left \lfloor \frac{2 - 3x}{x + 3} + x^2 \right \rfloor \)

f(1)  = 0

f(2)  = 3

f(3) = 7

f(4)  = 14

f(5)  = 23

f(6)  = 34

f(7) = 47

f(8)  = 62

f(9) = 78

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f (1000)  =  999,997

It appears that, from x = 3 to x = 8 inc;usive,  f(x)  =  x^2 - 2

And from x = 9  to x =1000 inclusive , f(x)  = x^2 - 3

So

So....the sum is

         8                      1000

3  +   Σ   x^2  - 2   +   Σ  x^2 - 3      =  333,830,510

       x = 3                 x = 9

x = 2x + 8                x = 13 -5x             x = 20 - 14x                     x = 40  + 5x

-x = 8                       6x = 13                 15x  = 20                         -4x  = 40

x  =  -8  NO             x = 13 / 6 OK          x = 20/15 = 4/3 NO         x = -40/4  = -10  NO

Only one value 

x = 13 / 6

floor (t)  =  -t^2 + 3t + 4

Here's a graphical solution  (subbing x for t )

https://web2.0calc.com/questions/help-stuck

After n steps, he has visited (n + 1)^2  lattice points (points with integer coordinates)

So

(2 + 1)^2  =  3^2  =  9

When Dirk finishes, Edith has run 14km  and Foley has run 9 km

So Foley runs  9/14  as fast as Edith

When Edith has run 24km, Foley has run  24 (9/14)  ≈  15.43  km

So Foley is   24 - 15.43  ≈  8.57 km behind

The system will have no solutions when the determinant =  0

1   1   3    1    1

1 -2    8    1  -2

k   0   5    k    0

Det  =  [ 1 * -2 * 5   + 1 * 8 * k ] - [ 3*-2 * k  + 5 * 1 * 1 ] =  0

[ -10 + 8k ] - [ -6k + 5 ] = 0

-15 + 14k =  0

14k  =  15

k = 15/14

-5s^2 - 16s + 39

This is a parabola that turns downward....no smalest value is possible