CPhill

Yes....Melody's answer is  correct.....I didn't account for the  fact that we want the ratio of even palindormes 4-digit palindromes to ALL 4-digit  integers... 

Thanks, Melody!!!!

Here's another way to solve this......

The area  of the triangle  =   

Integer coordinates on the boundary / 2  +  integer coordinates in the interior   - 1

The area  of the  triangle  is   20 * 5 / 2 =    50   units ^2

The line  joining   (-10,0)  and (0, 5)   has the equation     y  = x/2  + 5

So we will have integer  coodinates on this line when x=  -10, x = -8, x = -6,  x = -4,  x  = -2 and  x  = 0

Using symmety....we will also have integer coordinates when  x = 2, x = 4 ,  x = 6  x  = 8  and x  = 10

And we have  19 additional  integer coordinates on the  base

So.....the number of  integer coordinates on the  boundary  = 11 + 19  = 30

So

50 =  30 / 2  +   integer  coordinates in the interior  - 1

50  = 30/2  +  integer coordinates in the interior   - 1

50  =  15  - 1   + integer coordinates in the  interior

50 =  14   +  integer coodinates in the interior

36   =    integer coodinates in the interior

Even plaindromes

Starts  with                        Ends with

    2                                         2                 

    4                                         4

    6                                         6

    8                                         8

We have 10  choices for the middle two numbers  in each case   = 40  possibiliities

We have   90   4 digit palindromes   (the other 50  come from starting and ending the number with 1,3,5,7, or 9....and for each of these we have  10 possibiliites)

So....the  probability of an even palindrome  =   40  / 90    = 4 / 9

A 3rd degree polynomial has the  form      ax^3  + bx^2 + cx + d

Since  f(0)  = 0     then   x = 0   is one of the y intercepts  and    d  = 0

So....we can solve  for  a, b and  c     as follows :

a(-1)^3  + b(-1)^2  + c(-1) =  15  ⇒      -a   +  b - c  = 15     (1)

a(1)^3  + b(1)^2  + c(1)=  -5  ⇒              a    +  b  + c =  -5   (2)

a(2)^3  + b(2)^2  + c(2)  =  12  ⇒           8a   + 4b  + 2c  =  12    (3)

Adding  (1)  and (2)   we have that     2b  =  10 ⇒    b  = 5

Sub this into  each equation for b and we  have that

-a +5 - c  =  15 ⇒    -a  - c =  10    (4)

a +5  + c  = -5  ⇒   a  + c  = -10        (5)

8a + 4(5) + 2c =  12  ⇒   8a  + 2c = -8   (6)

Multiply (5)  by -2  and add it to (6)   and we get that     6a  =12 ⇒   a  = 2

Using (5)  to find c  we have that    2 + c  = -10 ⇒    c  =-12

So....the polynomial is   f(x)  =   2x^3 + 5x^2  - 12x      set to 0 and factor and we  get  that

(2x  - 3) (x + 4 )  = 0

Setting each factor to  0 and  solving  for  x  and  we  get the  other  two x intercepts  of   x  = 3/2    and x =  -4

See the graph here  : https://www.desmos.com/calculator/ef2xuaesrm

Let the small size cost  = C   and let the  volume of the  small  size be V

So....the unit cost for the  small size is  just  cost / volume  =   C/ V

The medium size  costs 50% more than the  small  size....so it  cost is  1.5C

The large size holds twice as  much as the  small size =  2V   and  costs 30% more than the medium size = (1.5)C (1.3)  = 1.95C

So...the unit  cost  for the large  size =  1.95C / 2V =   =   (195/200)C/V  =  (39/40)C/V =  .975C/V

Finally....the  medium size contains 20% less detergent than the large  size=   2V (.80)   = 1.6V

So....the uint  cost  for the  meduim  size  is  just   ( 1.5)C / (1.6 V) = (15/16)C/V  = .9375C/V

So.....best to worse we have

.9375C/V       .975C/V           C / V 

    M                   L                    S          ⇒        "E"

z^2  =  2i

(a + bi)^2  =   0 + 2i

a^2  + 2abi  - b^2   =   0 + 2i              

(a^2 - b^2)   + 2abi  =  0 + 2i        equate  coefficients

a^2 - b^2  = 0        (1)                     and             ab   = 1         (2)

(a - b) ( a + b)  =0 

(a - b)  = 0     or  ( a + b)  = 0

a = b          or         a  =  - b      (reject     because  a and  b will  have opposite signs and   ab  will be  negative if this is true  )

If   a  = b    then    (2)  becomes     a^2  =1        ⇒  a  = ±1

And if a   = 1   then b  = 1

And if a  = -1, then b  =-1

So....the solutions are       a  = 1, b  = 1     ⇒     z  = 1  + i        or    a =- 1 , b = -1   ⇒   z  = -1 - i 

We can geneate  a  liunear regression  approximation   using this website  : :   https://www.socscistatistics.com/tests/regression/default.aspx

The equation is

y = 1.58095 x - 1.12975

When x  = -3.....the approximate value fo y  is   -5.8726  ≈  -5.9

When x  = -20    .....the  approximate  value  for  y   is   -32.7488  ≈  -32.7

f ( √ [ x + 1) )   =  1  / x

f(2)   implies that

2  = √ [x + 1 ]        square both sides

4  = x + 1      subtract 1 from  both sides

3  = x

So

f(2)    =     1 / 3

We have a vertical asymptote at  x  = -1   because this value makes the  denominator undefined....

So....the  domain  iss (-inf, -1 ) U (-1, inf)

See my previous answer  as to the range of this rational function for the  graph

f(x)   =  3x - 7

            ______

              x +  1

We have  vertical  asymptote at  x = -1      and a horizontal asymptote at  y =  3

So......when x  < -1      y  is  >  3

And when  x  > -1       y is < 3

So....the range  is   ( -infinity , 3)  U (3 , inf)

Here is a graph : https://www.desmos.com/calculator/r9bpootve2