CPhill

A     3    E    4    B

                F             

C                      D

[EAD] = (1/2) (3/7)AB * AC

9 = (3/14)AB * 10

9 = (30/14)AB

9 = (15/7)AB

9 ( 7/15)  = AB

(63/15) = AB  =   21/5  

[ABCD ]  =  AB * AC  =  (21/5) (10)  =  42

(-4,0) and (0,2)  are on the  graph

The slope is  [ 2 -0 ]/ [ 0  - -4 ] =  2/4 = 1/2

Using the slope and (0,2)  the equation of the line is

y  = (1/2)x + 2         multiply through by 2

2y = x  + 4

Rearrange as

x - 2y =  -4

Angle BDC   = 30

To find base CD

CD / sin DBC  = BC / sin BDC

CD / sin 80  = 6 / sin 30

CD  = 6 * sin 80  / sin 30  =  6 * sin 80 / (1/2)  =  12sin(80)

Area of triangle BDC =

(1/2) BC * CD * sin 70  =

(1/2) 6 * 12sin 80 * sin70 =

36sin 80 * sin 70  ≈  33.3

To find BD

BD /sin70 = BC / sin30

BD /sin70 = 6 / (1/2)

BD / sin70 = 12

BD = 12sin70

Area of triangle ABD

(1/2) AB *BD sin ABD =

(1/2) *3 * 12sin (70)*sin (30)  =

18 sin (70) * (1/2) =

9 sin (70)  ≈ 8.46

Area of trapezoid  =  Area of triangle BDC + Area of triangle ABD  = 

33.3  + 8.46 ≈  41.76

60pi = pi * r^2 * (83/360)

60 (360/ 83)  = r^2

r  = sqrt [ 60 * 360  / 83  ] ≈  16.13

Not enough info to solve this....

\(\lfloor t\rfloor = 3t + 4 - \lfloor 2t \rfloor \)

See the graph here 

blue = floor (t)

green  = 3t + 4 - floor (2t)

Note that the graphs never intersect so there are no  solutions for  t

x^2 - 4x + y^2 - 12y = 0

x^2 - 4x + 4  + y^2 - 12y + 36 =  40

(x - 2)^2  + (y - 6)^2  =  40

This is  a circle  centered at ( 2,6)   with a radius of sqrt (40)

The max lies on the intersection of the line y = x + 4   and the  circle

x + y   max  =    [2 + sqrt (40)cos (45°)]  + [6 +  sqrt (40)sin(45°)]  =

2 + 2sqrt (10) * (1/sqrt 2)   +  6 + 2sqrt (10) * (1/sqrt 2)  =

[2 + 2sqrt (5)]  +    [6 + 2sqrt (5)]  =

8 + 4sqrt (5)  =

4 (2 + sqrt (5) )

Slope  =  [ b^2 - a^2 ] / [ b - a ] =  [ (b + a) (a - b) ] / [(a - b)] =  2

So

a + b =  2

a^2 / b  +  b^2 / a  =     [ a^3 + b^3 ] / [ab ] ^2  =   [(a + b)  (a^2  + b^2  - ab ) ] /  [ab]

x^2 -25x + 80  = -28x +  75       simplify

x^2 + 3x + 5 =  0

We have the form  mx^2 + nx + c =  0

Sum of roots

a + b =  -n/m =  -3/1 =  -3

Product of  roots  =  ab =  c / m =  5/1 =  5

a + b =   -3       square both sides

a^2 + 2ab + b^2 =  9

a^2  +2(5) + b^2  = 9

a^2 + 10 + b^2 = 9

a^2 + b^2  =  -1

So

[(a + b)  (a^2 + b^2 - ab) ] /  [ab]  =

[ ( -3)( -1 - 5) ]   / [5] =

[ 18 ] / [5 ]   =  

18 / 5 =  3.6

Let G be the total  number of  games   you  play

Let .75 G be the number of  games  you  have played

Let . 35 (.75)G be the number of  games  you  have won

Let .25G be the games  left

Let  .50G = the number of  wins  you  must have

And number of wins you  need - Number of  wins  you  have =  No. of  wins you need in  your remaining  schedule

So

[ No. of wins you  need - No, of  wins you have  ] / [ No. of Games Left ]

[ .50G - .35 (.75)G ] /  .25G  =

(.2375 G )  / ( .25G) = .95  = 95%    (you need to  win  95%  of your remaining  games )