ElectricPavlov

(3x-9)(x-2) = 0

3x^2 -15x + 18 = 0      m = 9

(3x-6)(x-3) = 0

3x^2 -15x + 18 = 0      m = 9 

You need 100   ODD numbers    1 - 199

1 3 5 7 9     and then 2-digit ones 11-99  then 3 digit ones 101-199

5                                         90                                    150 = 145 digits    (I think !)  

2 l + 2w = 32

l * w = 56      l = 56/w

Sub second into first equation

112/w + 2w = 32      multiply by w

2w^2 - 32w + 112 = 0      Quadratic formula shows  w = 8+ 2 sqrt 2        then  l = 8-2sqrt2

d^2 = (8+ 2 sqrt2)^2  + (8-2 sqrt2)^2

d^2 = 144      d = 12

Here is the Venn diagram

I would suggest a Venn diagram...I will post one in just a second...BUT you Q does not state how many dogs are in the school..... 

   Draw the three overlapping circles....

      then start where they all overlap   14 have all 3 ribbons

         then fill in the 2 circle overlaps with the info given.....

             then fill in the big circle portions with the info given.....

                 add up all of the numbers to find the number of dogs who learned a trick and got at least one ribbon

                           total 98 dogs

                                if there is more than 98 dogs in the class, the remainder learned no tricks....

l * w = 120     w = 120/l

l^2 + w^2 = 17^2 = 289

Substitute in the value of w

l^2 + 120^2/l^2 = 289     Multiply both sides by l^2

l^4 + -289 l^2 + 14400  = 0        let x = l^2

x^2 - 289x + 14400 =0       Quadratic formula reveals  x = 225  or 64

then l^2 = 225   or 64

        l = 15   or 8

         then w = 8 or 15       l x w  = 8x15=120    or   l x w = 15 x 8 = 120   

perimeter = 2l+ 2w = 2(15) + 2(8) = 46

Log both sx

(x+1) log 3 = log 4

(x+1) = log 4 / log 3 = 1.2619

x = .2619

then    3^(2(.2619)-1) = .5926

or another way:

3 * 3^x = 4

3^x = 4/3

3^(2x-1)  = 3^x 3^x   /3  = (4/3  * 4/3   ) / 3 =  16/9  / 3 = 16/27 = .5926

v0=5.56 m/s

vf=0  (when it hits the ground)     xo = original height   xf = final height = 0

a= -9.81 m/s^2

xf = x0 + v0 t + 1/2 a t^2

o = 108 + 5.56 t + 1/2 (-9.81) t ^2

0= 108  + 5.56 t - 4.905 t^2                          solve for t = 5.29 s

    1011

  11001

_____

100100

Great!  I got the same answer, but I like your method better !