CPhill

      30   A

12        X

            4

      90  C               B  60

The triangle is 30 - 60 - 90

AC = 12    BC =  12/ sqrt (3) = 4sqrt (3)     AB  = 8 sqrt (3)

Area of triangle ABC  =  (1/2) BC * AC  =  (1/2) (4sqrt (3) * 12 =  24 sqrt (3)

XC =  4

Area of triangle  BXC =  (1/2) ( BC) (XC)  = (1/2)(4sqrt (3)) ( 4)  = 8 sqrt (3)

Area of  triangle BXA =  [24sqrt (3)  - 8 sqrt (3) ]  =  16sqrt (3)

x ( 3x - 4)  ≤ [ 6x^2 - 3x + 5 ] /  5      multiply both sides  by 5

5x ( 3x - 4) ≤ 6x^2 - 3x + 5

15x^2 - 20x ≤ 6x^2 - 3x + 5

9x^2 -17x - 5 ≤ 0

Set up as an equality......this  is a  parabola that turns upward......the solution will be found on the  interval  that includes  the  roots

9x^2 -17x  - 5 =  0

The roots are

[ 17 - sqrt (289 + 160) ] / 18 =    [ 17 - sqrt (469) ] / 18  

And   [ 17 + sqrt (469) ]  /18

The solution is

x =  [  (17 - sqrt (469) )   /18 ,  (17 + sqrt (469))  / 18 ] 

Side of the square =   5/sqrt 2  in 

Area of the square = (5/sqrt 2)^2  = 25/2 = 12.5  in^2 

Area of circle = pi (diameter / 2)^2 = pi (6/2)^2  = 9 pi in^2 ≈  28.27 in^2

Area difference   =  [28.27 - 12.5 ] in^2  ≈  15.8 in^2

16 + 16x = 144           subtract 16 from both sides

16x  =  128               divide both sides  by 16

x = 128 / 16  =   8 oz

arc AQB =  (1/4) 2pi * 15  =  7.5 pi

PR  is a  diagonal of the rectangle = QO = the radius of the  quarter-circle = 15 

AP =  15  - PO

BR  = 15 -  RO

So

AP + BR  =   (30) - (PO + RO) 

But PO + RO  = (1/2) the perimeter of the rectangle = 17.5

So  AP + BR  =  30 - 17.5  =  12.5

Total perimeter of the  shaded area = 

7.5 pi  + 15 + 12.5  = 

7.5 pi + 27.5  ≈  51.06

87 kg  x     1 lb / .45 kg  =  ( 87/.45 ) lb ≈  193 lbs.

x^2+y^2 - 7 = 4y-14x+2x-6y+3

Rearrange as

x^2 +12x + y^2 + 2y = 10         complete te square on x,y

x^2 + 12x +  36  + y^2 + 2y + 1 =  10 + 36 + 1

(x + 6)^2 + ( y + 1)^2  = 47

This is a circle centered at ( -6, -1)  with a radius of  sqrt (47)

The area of the circle =  pi * r^2  =   pi * ( sqrt 47)^2  =  47 pi

Rearrange as

x^2 - 10x + y^2 + 10y = -c

Complete the square on x, y

x^2 -10x + 25  + y^2 + 10y + 25  = -c + 25 + 25

(x -5)^2  + (y + 5)^2  =  50 - c

To have a radius  of 1

50 - c   = 1

50  - 1 =  c

49  = c

\(\frac{a^2}{b} + \frac{b^2}{a} \)

Note that   the above can be simplified as   

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

Simplify the equation as

x^2 + 3x + 5 =  0

The sum of the roots, a + b =  -3      square both sides

a^2  + 2ab  + b^2  =  9         (2)

The product of the roots = ab = 5

2ab = 10

Sub this into  (2)

a^2  + 10  + b^2  = 9

a^2 + b^2  =  -1

Sub all this into (1)

[ (-3) ( -1 -5) ] / [5]  = 

[ (-3)(-6) ] / [5]

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