CPhill

Here's my best attempt  (I'm not great at these counting problems !!! )

Non-prime digits = (0, 1, 4, 6, 8, 9 )

Number of three digit numbers not having any prime digits

Hundreds place  = (Choose  any  one of the digits 1,4,6,8,9) =  C(5,1) = 5

Tens place  = ( Choose any one of the digits 0,1,4,6,8,9) = C(6,1)  = 6

Ones place = (Choose any one  of the digits 0,1,4,6,8,9) = C(6,1)  = 6

So   5 * 6 * 6 =    180 have no prime digits

So

900  three digit numbers -  180 with no prime digits  =    

720  have at least one prime digit

f (f(x)) = 3(3x - 2) - 2 =  9x - 8

f (f(f(x)))  = f (9x  -8)  =  3(9x - 8) - 2  =  27x - 26

f(f(f(f(x)))) =  f(27x - 26)  =  3 (27x  -26)  - 2 =   81x - 80  =  g(x)

g ( 1)  =   81(1) - 80  =  1

g(2) =  81(2)  -  80  =   82

Domain  of g  =  [ 1 , 82 ] 

Solve this

x^2  = 8

x = sqrt (8)  = 2sqrt (2)     or   x = -sqrt(8)  =  -2sqrt (2)

The width of the interval   is     2sqrt (2)  -   - 2sqrt (2)  =   4sqrt(2)

THX, BuilderBoi.....I didn't see the "+" sign  !!!

We have the form

y = a ( x -h)^2  + k          where (h,k)  = the vertex  and  (x,y) is a point on the parabola

So

175 = a ( 1 - - 4)^2 +  0

175 =  a (5)^2 

175  = 25a

175 / 25  = a =   7

Note that  x^2  - 4x - 5 =  ( x - 5) ( x + 1)

Multiply through by these two factors and we  get

12x + 22  =   A( x + 1)  + B ( x -5)       simplify

12x + 22 =  (A + B)x  +  ( A - 5B)        equating terms we have

A + B  = 12     →   B = 12 -  A     (1)

A - 5B  =  22       (2)

Sub (1) into (2)

A - 5 ( 12 - A) = 22

A - 60 + 5A  = 22

6A  = 82

A =  82/6  =  41/3

And   

B   = 12  - 41/3  =   36/3 - 41/3   =  - 5 / 3

A + B  =   36/3  =  12

Three sides of the rectangle (a square) will be fenced  = 3*35  =  105ft

The semi-circle will have a radius of   (35/2)  =17.5 ft

And the additional fencing required = (1/2) the perimeter of the circle  = pi (17.5)  = 17.5 pi ft

So.....the total fencing required =  105 + 17.5 pi ≈  160 ft

The radius^2  =   (PQ)^2  =  ( 9 - - 3)^2  +  ( 4 - -3)^2  =  12^2 + 7^2  =  193

The area =     pi * 193  =

193 pi

 Number of terms in the series =  (104 - 2)  / 2  + 1  = 51 + 1  =    52

Sum of the series  =   

(2 + 104) * (52)  / 2  =

53 * 52  =

2756

2756 mod 7  =   5

f(-2) =  (-2)^2 + 3 =   7

t( f(-2)) =   t (7)  =  9 + 2f (-7) =  9 + 2 (7^2 + 3) =  9 + 2( 52)  =  113