ElectricPavlov

(NOT SURE OF THIS ANSWER)

  Ways to choose 2 bracelets    = 6

    ways to choose who gets 2 bracelets = 4

        the other three split the remaining 2 braclets   ...I'll count 'no bracelet ' as a color option    so three bracelets to distribute to 3 people

                = 6  

6 x 4 x 6 = 144 ways  

Multiple ways to solve this

  rememebr   tan = sin / cos       you are given cos y      this is also  sin x

           find cos x          via     cos ^2 + sin^2 = 1

                                                 cos^2 = 1 - sin^2

                                                  cos^2 = 1 - (7/29)^2 = 1 - 49/841 = 792/841

                                                    cos x = sqrt (792)/29 =   6 sqrt 22 /29

tan x = sin x / cos x =  6 sqrt 22 / 29   /  7/29 =  6 sqrt 22 / 7

Remember in a rt triangle   sin = opp side / hypotenuse

we will need to find side DC 

Pythag theorem    DC = sqrt (9^2 - 5^2) = sqrt 56

sin b =   sqrt 56  /  9 = 2 sqrt 14 / 9       = .83148   

......or you can find cos b = 5/9     then find   sinb   with the trig identity    cos^2 + sin^2 = 1

                                                                                                                      25/81  + sin^2 = 1

                                                                                                                          sin^2 = 1- 25/81

                                                                                                                                   = 56/81

                                                                                                                            sin = sqrt 56 / 9 =  2 sqrt 14 / 9        as before.....

Here is a desmos picture:

Here is a second way:

the two given points are equidistant from the center    (x, 0)

Use distance formulas for the two points and the center

(x-0)^2  + 4^2 = (x-1)^2 + 8^2

   solve for x = 49/2      so circle center is   49/2 , 0

use one pont to solve for r in the circle formula   (0,4 )

( 0 - 49/2)^2 + (4-0)^2 = r^2

   r^2 = 616.25                       r = sqrt (616.25)

*** edited ***   Equate the two parabolas and solve for x to find the x coordinates of where they intersect

x^2 + 4x + 6     =    1/2 x^2 + x + 4

1/2 x^2 + 3x + 2 = 0        use quadratic formula to find    x=−3 +- sqrt5   <===== this is a and c

    you should be able to finish this......   

For x , the midpoint will be   exactly between 1 and -10   at   - 4.5

  for y   the midpoint is exacly between 4 and -17   at  - 6.5      You can finish ....

if all 0's count as one of the numbers there will be 10¹³  possibles :   10 000 000 000 000     (ten trillion)

We need to move from point Q the same distance as from P to Q

  for 'x'  moving from P to Q we go from 5 to 3     or  -2      go  -2 MORE from Q ....from    3 to 1     

    for 'y' moving from 3  to  8    is  + 5     move + 5 more from Q   to 13

Point T is then   1,13

First ....graph the lines ..then when you find thepoint of intersection, you can calculate the area of the rectangle (below) and subtract the area of triangle 1 and 2 :