ElectricPavlov

8/125 ^x = 5/2       LOG both sides to get 

x LOG (8/125 ) = LOG (5/2) 

x =LOG(5/2) /  LOG (8/125) 

x =  - 1/3  

(8/125) ^-1/3  =  5/2 

f = fish #

5/8 f = shell number 

(5/8f -50) / (f-8)   = 1/4        <====  solve for f ....cross multiply to get:

  4 ( 5/8f-50) = f-8 

20/8f - 200 = f-8 

12/8 f = 192 

f = 128 

then ORIGINAL shells were   5/8 (128) = 80 

   FINAL shells = 80 - 50 = 30 shells in the end 

Denominators cannot equal zero 

   so x obviously cannot   = 0      in 3/x          <=====first one 

     then   1 + 3/x   cannot equal zero....so 'x' cannot =  -3               <====second one   

         then   1 + 2/(1 + 3/x)   cannot equal zero....this will happen if  x = -1                   <====third one 

Sum total  :     0 + -3 + -1   = -4   

3x -7   in the required interval  can only be   (2  thru   6 1/3 )

   then 2x    to be in the interval can only be  (-1/2 through 6)         so     the common factors are  x =   (2,6)      

Pick any two points on the line to calculate the slope = rise/run = 1/2 

    then    y = 1/2 x + b      sub in any point to calculate the value of 'b'

               2 = 1/2(0) + b     shows b = 2

 y = 1/2 x + 2    re-arrange

1/2 x - y = -2     multip;ly through by 2  ( to get rid of the fraction)

x - 2y = -4 

y-axis intercept occurs when x = 0 

    3(0) -6y = 12 + 2(0) - 4y 

          -2y = 12 

             y = -6                y intercept = (0,-6) 

x axis intercept occurs when y = 0 

   3x - 6(0) = 12 + 2x - 4(0) 

       x = 12                      x intercept = (12, 0 )

Using Pythagorean theorem    d^2 = (-6)^2 + 12^2 

                                                 d = sqrt (180)  = 6 sqrt (5) 

52 P 5  =  52! / 47! = 311 875 200 ways to pick 5 cards                ( 47  is   52 - 5 ) 

= x^2 -x -1 = 0     Quadratic Formula   with    a = 1     b = -1      c = -1 

         to find the x values are    1/2 ± sqrt(5) / 2

Only in Q I   nad Q III   is tangent a positive number      ( remember tan = sin/cos) 

    since the question asks for Φ between  0 and 180 degrees, you only get one answer in Q I

        using a calculator         arctan (9) = 83.7 degrees 

Total area = pi r^2 + (2r * 2r)      ( rememeber the sides of the square = 2r )

The area =  pi  r^2  +  (2r)^2   =  (pi +4) r^2  =  7.142 r^2

    Then  solving for 'r'  =====>   sqrt (area/7.142) = r

                  First one:  area = 9

                                      r = sqrt ( 9 / 7.142) = 1.1  in

                                         square side = 2r = 2.2 in         ( rounded answers)

                 second one:  area 20 in^2

                               r = sqrt ( 20 / 7.142) =.......     etc

I'll let you do the others...