CPhill

Teams  left after round r  =  128  / 2^r        

128 / 2^4  = 128 / 16 =   8   teams  left

I'm not sure  of the notation, but I believe that the first  is a 180°  rotation about the origin

180° rotation    (x,y) → (-x, -y)

New  coordinates  X =  (4,-7)   Y =  (0, -8)  Z  =( -2, 1)

Second one is a  270°  rotation about the origin  = -90° rotation about the origin

-90° rotation   (x, y)  =  (-y, x)

New coordinates    X =  (-7, -4)   Y = (-8,0)  Z = ( 1, 2)

Simplify as

-5s^2  - 16s   + 39

This is a parbola that turns  downward.....there is  no  min

-8 ≤ x + 17               x + 17 <  -3 - 5x

-25 ≤ x                     20 <  - 6x

                                20/-6 > x

                                   -10/3 > x

[ 25 , -10/3 )

sin 62°  =   height of balloon / length of rope

sin 62° =  height of balloon /  30 m

(30 m) ( sin 62°)  = height of balloon  ≈  26.49 m ≈  2649 cm

We can solve these equations

7 = a (-3)^2  + b(-3)  +  c

5 = a(-1)^2   + b(-1) +  c

8 = a(2)^2    + b(2)  +  c

Simplify  as

9a - 3b + c   =  7         (1)

a  - b  + c  =  5        (2)

4a + 2b + c  =  8         (3)

Multiply  (2)  by -3 and add to (1)

6a - 2c  = -8   →   3a - c  = -4    (3)

Multiply  (2) by 2  and add to (3)

6a  + 3c  = 18   → 2a + c = 6     (4)

Add (3)  and (4)

5a = 2

a = 2/5

To find c

2(2/5) + c =  6

4/5 + c  = 6

c = 6 -4/5   =  26/5

To find b

2/5  - b + 26/5 =  5

b  =  2/5 + 26/5 - 25/5

b = 3/5

a + b +  c  =   

2/5 + 3/5 + 26/5    =

31 / 5

x = -8  is  one  x intercept

The other x intercept is

x =  3 + (3 - - 8)  = 3  + 11  =  14

So  we have that

P(x)  =a ( x + 8) ( x -14) 

If the y intercept is 5, then the point (0,5)  is on the graph....so.....

5 =  a(0 + 8) ( 0 -14)

5 =  (8)(-14)a

5 =  -112 a

a = -5/112

P(x)  = (-5/112)(x + 8) ( x -14)

Here's the graph :

Simplify  as

x^2 - 16x + y^2 + 22y  =  80   complete the square on x, y

x^2 -16x + 64  + y^2 + 22y + 121  =   80 + 64 + 121

(x - 8)^2   +   (y + 11)^2   =  265

The center is (8,  -11)     and the area  is   265 pi ≈  832.5

A

4             5   

B       D          C

          3

Let DC  = x  and  BD  = 3 - x

Since AD is an angle bisector

DC / AC = BD / AB

x / 5  = (3 -x) / 4        cross-multiply

5 (3 - x)  = 4x  

15 - 5x  = 4x

15  = 9x

x = 15/9  = 5/3  = DC  = the base of [A

And the height of  triagle ADC =  AB =  4

So

[ADC ] =  (1/2) DC * AB =   (1/2)(5/3)(4)  =  20 / 6  =  10 / 3 =    3 (to the nearest integer)

\(1 + \frac{3}{6} + \frac{5}{6^2} + \frac{7}{6^3} \)

9/6  +   5/36  + 7/216

[ 324 + 30 + 7 ] / 216

[ 361 ] / [ 216]