CPhill

3z + 4iz'  = 1 - 8i

3 (a + bi) + 4i(a - bi)   = 1 - 8i

3a + 3bi + 4ai + 4b  = 1 - 8i

3a+ 4b =  1      ⇒   12a + 16b  = 4

4a + 3b  = - 8 ⇒     -12a - 9b = 24

7b  = 28

b = 4        and   3a + 4(4) = 1   ⇒  3a = - 15  ⇒  a = -5

So

z = -5 + 4i

Let a/2  = B      so a  = 2B

And we have the identity that

cos 2B  = 2cos^2 B - 1        add 1 to both sides

cos 2B + 1  = 2 cos^2 B        divide both sides by 2

cos 2B + 1

________    =  cos^2 B       take both roots

     2

±√ [ ( cos 2B + 1 ) / 2 ]  = cos B

So.....

True  !!!

cos 2a  =  cos a * cos a -  sin a  * sin a   =   

cos^2a - sin^2a  = (1 - sin^2 a) - sin^2 a  = 1 - 2sin^2 a

So...

True   !!!!

Find the intersection points

2x^2  = 12 - x^2

3x^2  = 12

x^2  = 4

x  = - 2  and x  = 2

So we have, using symmetry

    2                                              2                                                 2

2 * ∫  12 - x^2 - 2x^2 dx  =   2    ∫  12 - 3x^2  dx   =  2 [ 12x - x^3]      =  2 [ 12(2) - (2)^3 ]  =

    0                                              0                                                 0

2 [ 24 - 8]  =  

2 * 16  =

32 units^2

The form of the circle is

(x - 5)^2 + (y - 15)^2 = 130

Since the x coordinate is twice the y coordinate, we have

(2y - 5)^2 + (y - 15)^2  = 130      simplify

4y^2 - 20y + 25  + y^2 - 30y + 225  = 130

5y^2 - 50y  + 120  = 0   

y^2 - 10y + 24  = 0

(y - 6) (y - 4)  = 0      set each factor to 0 and solve for y  and we get that  y = 6  and y  = 4

So....the max value for x  = 2y  =  2(6)  =  12

  7

Andy ate  4/7  = 84/168  pie

Barry ate 7/12  =  98/168 pie

Craig ate 5/8  = 105/168 pies

Andy's pie remaining = (1 - 4/7)  = 3/7

Barry's pie remaining = (1 - 7/12)  = 5/12

Craig's pie remaining = (1 - 5/8)  =  3/8

So  Fido  ate   3/7 + 5/12 + 3/8   =  205/168  pies

So Fido ate   205/168 - 105/168  = 100/168  = 25 / 42  more pies than the winner

We have that either

x + 4 + 3x > x + 9         or      3x + x + 9 > x + 4        or      x + 4 + x + 9 > 3x

4x + 4 > x + 9                           4x + 9 > x + 4                       2x + 13 > 3x

3x > 5                                        3x > -5                                    13 > x        

x > 5/3                                        x > -5/3                                   x < 13

Taking the most restrictive interval

5/3 < x < 13

So

n - m      =    13  - 5/3   =   39/3  - 5/3   =    34 / 3

Nice, tertre!!!!!

No clue, Jenny....did your instructor point you to any technology that might be used to do this???.....

x^2 - 6x  = 8      complete the square on x

x^2 - 6x + 9  =  8 + 9

(x - 3)^2  =  17     take both roots

x -  3  =   ±√17      add 3 to both sides

x  = 3 ±√17