CPhill

Simplify as

8x^2 - 11x - 45 =  0 

Quadratic Formula

x =  [ 11 ±  sqrt  [ 11^2  - 4 (8) (-45) ] ] /  [2 * 8 ]  = 

[ 11 ±  sqrt [  121 + 1440 ] ]  / [16 ]  =

[ 11± sqrt  [ 1561 ] ]  /  16  

Changing angle XAY to  16°

Then angle  AYX  = angle  AYB =  measure (AXY) - measure (XAY ) =    90°  - 16°  =  74°

J /  S = 3/2

J =  (3/2)S

( J + 12 )  / ( S  - 1)  =   2/1

((3/2)S+ 12) /  (  S  -- 1)    =  2/1               cross-multiply

(3/2)S + 12  = 2 (S - 1)

(3/2)S + 12  = 2S   - 2

12 + 2 =  2S - (3/2)S

14 =  (1/2)S

14 ( 2/1)  = S

28  = S

J = (3/2)(28)  =  42

After changes    J + 12 =  54           S - 1 =  27

54 +  27   =    

81 total  students

When the perimeter is doubled  the scale factor  =  2

Original area  =   New Area /  Scale Factor^2  =   3600 / 2^2  =  3600 / 4 =  900

ACB is  an inscribed angle in the  circle  which intercepts the same  arc as central angle AOB....ACB has half the measure of  AOB

So

ACB = (1/2)AOB

2 * ACB = AOB

2 * 40  = AOB  =  80 (degrees)

Thx, DS2011   !!!!!

Set the y's equal and  simplify as   

 2x^2 +  9x   - c =  0

-2x^2 - 9x + c  =  0

One real solution, discriminant =  0

9^2  - 4 (-2)(c)  = 0

81  + 8c =  0

81  =  -8c

c = - 81 / 8

No real solutions,  discriminant <  0

81 + 8c < 0

81 < -8c   →  -8c > 81                (divide by -8, reverse the inequality sign)

c < -81/8

More than one real solution   c >  -81/8  

c/ sin (105)  = 7/ sin (35)

c  =  7sin (105) / sin (35)  ≈  11.8

a / sin ( 40 )  = 7/sin (35)

a  =  7 sin (40) / sin (35) ≈ 7.8 

Semi-perimeter  of triangle  = (7 + 11.8 + 7.8) /  2  = 13.3 = S

Area of triangle  sqrt [ 13.3 (13.3- 11.8)(13.3 - 7.8) (13.3 - 7) ]   ≈  26.3 =  A

Incircle radius  =     A  /  S  =   26.3 / 13.3   ≈ 1.98

Circumcircle  radius  =   abc   /  ( 4A) =   (7 * 7.8 * 11.8)  /  [4 * 26.3 ]  ≈  6.12

x - y  =   pi ( 6.12^2 - 1.98^2)   ≈  105.35  

Two Secants Segments Theorem

3 (3 + 6)  =  4 ( 4 + x)

27 =  16  +   4x

11 =  4x

x = 11 / 4

AOB is a  central angle....it has the same  measure as its  intercepted arc