CPhill

We actually just have 4 circles each with a radius of 5  (diameter of 10)

Area =  4 * 3.14 * 5^2  =  100 (3.14)  = 314 cm^2

Perimeter  = perimeter of  8 circles + 8 diameters =   4 * 2 * (3.14) * 5  + 4 * 10  = 165.6 cm

(t - 6) (2t - 5)  = (2t - 9) (t - 5)

2t^2 - 17t + 30 =   2t^2 - 19t + 45       simplify as

2t = 15

t = 7.5

When the radius is doubled  the perimeter is also doubled

So

2pi * 4 -  2pi * 2 =   

8pi  - 4pi =

4 pi

Something like this  :

  W                               X

   Z    8     A          B  3  Y

If ZWX is trisected, then  angle  AWZ = 30  and  angle  BWZ  = 60

So triangle  AZW  is a 30-60-90 right triangle

So WZ = AZ * sqrt 3  =  8sqrt 3 

And triangle BZW is a 30 -60-90 right triangle

BZ   =   WZ* sqrt 3  =  8 sqrt 3 * sqrt 3  =  24

So   [ WXYZ ]  = [ WZ * ( BZ + BY) ] = [ 8sqrt (3) * ( 24 + 3) ] =   216sqrt (3) units^2

To make things easier....multiply all three sides by 3

Semiperimeter =  ( 4 + 6 + 7)  / 2 =    8.5

Area of  this triangle =   sqrt  [  8.5 (8.5 - 4) (8.5- 6) (8.5 - 7) ]  ≈   11.98

The scale factor of the given triangle to this one  =   1/3

The area of the given  triangle  = 11.98 * scale factor^2 =   11.98 * (1/3)^2  ≈    1.33 units^2

4x + ay  = 8

2x + y  =  b    

Multiply the second equation through by 2

4x + 2y  =  2b

To have infinite solutions to the system one equation must be a multiple of the other

a  = 2

And

2b = 8

b =  4

So

(a , b)  =  ( 2 , 4)

3/x + x   = b          multiply through by x   and rearrange

x^2  -bx + 3   = 0

If we have one solution then the discriminant =   0   .....so.....

b^2  -  4*3*1  = 0

b^2  - 12  = 0

b =  sqrt (12)  =   2sqrt (3)

LOL!!!

Thx....I just get lucky once in a while  !!!!!

(1/2) of the gray area  =

Area of a sector of 1/3 of the circle  - area of an isosceles triangles with equal sides of 10 and  an included angle of 120°  =

(1/3) pi * 10^2  -  (1/2) 10^2  sqrt (3) / 2 =

100  [  pi/3  -  sqrt (3) / 4 ] 

So....the whole gray area  = 

200 [ pi/3 - sqrt (3) / 4 ]  =  200 [ 4pi - 3sqrt (3) ] / 12 = (50/3) ( 4pi - 3sqrt (3))  units^2 =

 ≈   122.84 units^2

Call the total Number N

P + I  = (9/16)N   →   (55 - D) + I  = (9/16)N      (1)  

D + I =  (3/4)N             (2)

Add (1) , (2)

55 + I + I  =  (9/16)N + (3/4)N

55 + 2I  =   (21/16)N

2I  =  (21/16)N -  55

I =  (21/32)N - 55/2

So

I + P + D  =  N

(21/32)N  - 55/2 +  55  =  N        multiply through by  32

21N + 880  = 32N

880  =  11N

N =  80

I =  (21/32) *80  - 55/2 =     25

P +  I =  (9/16) (80) =   45

P = 20

I - P  =    

25  - 20  =   5 

Ivan had 5 more than Peter