CPhill

1)   7125   -  2200   =   4925 kg of non-plastics

Pct  of non-plastics =   4925  /  7125    ≈    69%

2)    To  see  this   let's  double the amt she bought and the amount the  cat ate

So  she buys 100 kg  and the  cat eats 22 kg

So    100 - 22  = 78 kg are left

So      78 /  100  =   78%  left

3)   5 games  of the total number   (T) =  12.5%  =  .125

So  

5 / T  =   .125     multiply   both sides by  T

5 =  .125 T       divide both sides by  .125

5 / .125 =  T  =    40 games total

He must  have won   40 T - 5 he missed  =  35

Since the right triangle is isosceles.....we need to first find the  equal legs of the right triangle

So we have...[by the Pythagorean Theorem ] 

L^2  + L^2  =  50

2L^2    =50

L^2   = 50  / 2

L^2  = 25        take the positive root

L =  5 cm =   the equal legs

The surface area  of  one of the right triangles = (1/2) ( product of the legs)   = (1/2)(5 * 5)  = 25/2  = 12.5 cm^2

But we have two of these  (at the bottom and top)  so the total surface area of top/bottom= 2*12.5 = 25cm^2

The area of  the sides  =  perimeter of the right triangle * height of prism

So   we  have  5 + 5 + sqrt 50  = 10 + sqrt 50  =  perimeter

So  area of sides =   [ 10 + sqrt 50 ] * 8  =  80 + 8 sqrt 50  ≈  136.57  cm^2

So  the total  surface area  =   [ 25  + 136.57 ] cm^2   =  161.57 cm^2   

2.      Note that we can express this in  a slightly different manner   :

We  can write

      6                                    6                                6                                        6                                                        

_____________   +    _____________   +     ______________   +  ... +  ____________

[2 (1) + 1]^2 - 1           [ 2(2) + 1]^2 -1              [ 2(3) + 1]^2 - 1                [2(n) + 1]^2 -1

Note  that

       6                           6                                  6                      3          [     1           1          ]

__________  =     ______________   =    __________  =   ____  *       ___  -   ____

(2n +1)^2 - 1         4n^2 + 4n + 1 - 1           4 (n^2 + n)            2          [     n         n + 1     ]

So  we  can write

(3/2)   [    1/ 1 -  1/ [ 1 + 1)   +    1/ 2  -  1 / (2 + 1)   +  1/3  -  1 ( 3 + 1) +  1/4 - 1/(4 + 1) +...+  1/n - 1/n+1)  ] =

(3/2)  [   1  - 1/2  + 1/2   - 1/3  + 1/3  - 1/4  + 1/4 - 1/5 + ..... + 1/n  - 1/(n+1)  ]   =  

(3/2)   [  1    -    1 / ( n + 1)  ]  

As  n approaches some  "large"  number,   1/ ( n + 1)     approaches  0

So.....the sum  is

(3/2)  [  1  +  0 ] =

(3/2) [ 1 ]  =

3/2

I think that we need more info

Area of a square =  side^2

225 ft^2    = side^2                take the positive square root of both sides

sqrt ( 225 ft^2)  =  sqrt  (side^2)

15  ft =  side

And the perimeter of a square  =  4*side  = 4*15  =   60  ft needed

9)   Again, we just have a counting problem which can be expressed  as a combo multiplication

Choose  1  of each color and we  have

8C1  *  4C1  *  3C1   =

  8    *    4     *  3   =

           96   ways

10 )     Obviously  an English book could come first or  a Math book could come first  = 2 ways

And for  each of these.....we can permute each type of book in 4!  ways

So     2 C1  *  4!  * 4!    =   2 * 24 * 24   =    1152  ways

Here's some

6)  This isn't a perm or a comb prob....it's just a counting prob

The number of possible meals =  3 * 4 * 2   =  24 possible meals

7)  The fathers grouped together can occupy 9 diiferent positions  and  for each of these they can be arranged in 4! ways.....and for each of these the mothers can be arranged in 8!  ways

Again.....this is really nothing more than  a counting problem.....the number of ways are

9 * 4! * 8!  =    8709120  ways

8)  Note that, since the O's are identical, there is only  1  identifiable way that they start and end a "word"

And  the other 4 letters  can be arranged  in        4!  / (2! * 2!) =  6 identifiable ways

So  1  *  6 =    6 identifiable ways        

Let's write

y = 16 / ( 5 + 3x)          the object is to first get x by itself.......multiply both sides by 5 + 3x

y ( 5 + 3x )  =  16        simplify

5y +  3xy  = 16           subtract  5y from both sides

3xy  =  16   - 5y            divide both sides  by  3y

x =   ( 16 - 5y)  / ( 3y)        "swap"  x and y

y = (16 -5x)  / ( 3x)  =   the inverse  =  f^-1(x)

f^-1(-2)   =  (16 - 5(-2) )  / (3 * -2)  =    26 / -6   =  -13/3

Check that  f(-13/3)  =    -2

Add the first two equations

ax + y  + x + ay   = 2      simplify

a ( x + y)  + (x + y)  =    2                                        { remember that  x + y  =  9 }

a (9)  + (9)  =  2

9a  =  2  - 9

9a  =  -7

a   = -7/9

See the graph here   :    https://www.desmos.com/calculator/1tn3bcfeqk

\($1 \ \square \ 2 \ \square \ 3 \ \square \ 4$\)

+ + +   =    10

+ + -    =      2

+ - -     =     -4

- - -      =     -8

- + +    =      6

- - +     =      0

+ - +    =      4

- + -     =     -2