CPhill

The hour hand moves at   (1/2)°  every minute

Let the minute hand  be at  0°  at 9:45

At this time the hour hand  would have moved   (45)(1/2)°  =   22.5°  since 9:00

So  the acute angle = 22.5°

Note that DC  = CE

And DC =  BC

So  BC  = CE

Angle  BCE =  angle BCD - angle  ECD =    90  -  60     =  30°

And triangle  CEB is isoceles  with  BC = CE

So angle CBE  = [180 - 30] / 2  =  150/2  =   75°

Note that  because  CM = MA   then angles  MCA and MAC  are equal......call their value x

And because of the Exterior Angle Theorem, angle BMC  = 2x

So

BMC + angle A  = 177

2x + x  =177

3x  =177

x =177/3   = 59°  = angle A   (and angle C )

So angle CMA  = 180  - 2(59)  = 180  - 118  =  62°

And since  BM = CM  then angles   CBM and BCM are equal...call their value y

And by the Exterior Angle Theorem

Angle CMA  = angle CBM  + angle BCM

62  =  y  +  y

62  =  2y

62 / 2  = y  = 31°  =  measure of angle CBM   (angle B )

Number of hours in 9 weeks =   9 * 40  = 360

Earnings per hour  =   4320  / 360   =  $12

C =  [  14 + 3(14-2)  ,  4 + 3(4 - - 2) ]  =   [  14 + 36 , 4 + 18 ] =   ( 50, 22)

Note that we can write    1/ [ n^2 -1]   as   1/[ 2(n -1)] - 1/[2(n + 1)]

So we have

6  (  1 / [ 2(3-1)] - 1/[2(3 + 1)]    +  1/[2 (4-1)] - 1 /[2(4 + 1)]  + 1/[2(5-1)] - 1/[2(5 + 1) ]   +  ..........  )   =

6  [ 1/4 - 1/8 + 1/6 -  1/10 + 1/8 - 1/12 + 1/10 - 1/14 + 1/12 - 1/16 + 1/14 - 1/18 + ......] 

All the terms in red will "cancel"  and we are  left with

6  [ 1/4 + 1/6]  =

6 [ 10 / 24 ]   =

6[ 5/12]  =

5 / 2

c(1)  = 3/16

c(2)   = c(1) * 4  = ( 3/16) * 4 =  12 /16 =  3/4

c(3)  = c(2) * 4   = (3/4) * 4   =   3

8x^2  + 15x + c  = 7x^2 + 11x      rearrange as

x^2  + 4x  +  c   =  0

If we have two real roots the discriminant  is >  0

So

4^2 - 4(1)(c)  > 0

16 >  4c

4 > c

c < 4

So  c =   1, 2  or 3 

The product of all possible c's  =  1 * 2 * 3   =  6

Just set  up   as coordinates in the real plane

(-19, 32)  = A

(-5, 12)  = C

(-22,15)=   B 

Note that from B  to C, x changes by  17   and y changes   by    -3

So the remaining coordinate =  (-19 + 17, 32 - 3) =   (-2, 29)   =  D

Expressing this in complex form   =  -2 + 29i

See  here  :

a = k / b   where k is the proportionality constant

-6   = k / -3

-6 * -3  =  k =   18

So when b = 8

a  = 18 / 8     =    9/4