CPhill

-2x -5y = -9       multiply by -3  → 6x + 15y = 27   (1)

-6x - 15y =-21    (2)

Add (1) and (2)  and we have

0 = 6    so......no solutions exist

Here : https://www.desmos.com/calculator/owddboe2e6

y  =  a      .....will always be horizonttal line at "a"

((sqrt(37)cos(t))/37)^2  ....we can express this as.....

(√37)²  * cos² (t) /  37²  =

( 37) / (37²) * cos² (t)  =

(1/37) cos² (t) =

cos² (t) / 37

18*200 = 24*V₂

3600 = 24V₂      divide both sides by 24

150  = V₂

4 + 4/5  =   24/5    ...so.....

(24/5) (24/5)  =  576 / 25

-39 < 6v + 9  ≤ 39         subtract 9 from everything

-48 < 6v  ≤ 30         divide by 6

- 8 < v ≤ 5

We are solving this :  [ the shot will be at 0 m above the ground at the finish ]

-0.004s^2 +0.64s+1.94  = 0   multiply   by  -250

s^2 - 160s -  485  = 0       add 485 to each side

s^2 - 160s   = 485 

Take 1/2 of 160  [80] .........   square it  [6400] and add to each side

s^2 - 160s  + 6400   = 485 + 6400        factor the left side and simplify the right

(s - 80)^2  = 6885        take the + √ root

s - 80  =   √6885       add 80 to boh sides

s =  √6885  + 80 ≈    162.98 m    [ that's one heck of a throw....LOL  !!!  ]

375x^3+24  =

3 [ 125x^3 + 8 ] =        factor as a sum of cubes

3 [ 5x + 2 ] [ 25x^2 - 10x + 4 ]

141 rooms....choose any 5 of them    = C(141,5)  = 432,295,143  ways

Sum of natural numbers < 400  =   399 * 400  / 2   = 79800

Sum of natural numbers  < 400 divisible by 5  =  5 [ 1 + 2 + 3 + .....+ 78 + 79  ]  =

5 * [79 * 80 / 2]  = 15800

So

Sum of natural numbers  < 400 not divisible by 5  = 79800 - 15800  =    64000