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EP is correct.

I used the same method as EP and found

As x increases by 2,   P(x) increases by 6

This is a linear relationship, hence it will be a line.   

The gradient will be  6/2 = 3

The y intercept is 2

so

the line is

P(x)=3x+2

Ok I solved the first part:

the digits 01 through 42 represent a response of nonfiction.

47 71 18 92 00 20 40 89 02 26 48 33 77 84 74

But can anyone help understand What does p and p̂ mean - in this context??

The altitude  of the  cone  =   7sqrt(3)   cm

The radius  of the  cone =  7  cm

The volume of the  cone  =

pi  ( 7)^2  ( 7sqrt(3) ) / 3    =   7^3 pi  / sqrt (3)   ≈    622  cm^3

Since  AC  = 12.....then  AD  = 6  = CD

And since AB  = 16, then AE   = 8  =  BE

And since triangle DAE  is right.....then   DE =sqrt  [ AD^2  + AE^2 ]  = sqrt [ 6^2 + 8^2 ] =  sqrt (100)  =10  = GF

And ABC  is a 12 - 16 -  20  right triangle

And   CG  + FB  = BC  - GF  =  20  - 10   =  10

Let  FB  = x     so   CG  = 10 - x

So

CD^2  - CG^2  = DG^2

And

BE^2  - FB^2  = FE^2

And  DG = FE   so 

6^2  - (10 - x)^2  =  8^2  - x^2      simplify

36  - x^2 + 20x -100  =  64  - x^2

-64 + 20x  =  64

20x  =  128

x  =  128 / 20   =  6.4

So  FB   = 6.4

And  

FE   =   sqrt  ( BE^2  - FB^2)  =   sqrt ( 8^2  - 6.4^2)  =   4.8

So   area  of  right triangle DAE  =  (1/2)(AD)(AE)  = (1/2)(6)(8) =  24      (1)

And the area  of DEFG =  (DE)(FE)  =  10 * 4.8  =  48     (2)

So.....[  AEFGD ]  =     (1)  + (2)   =     72 units^2 

Third one

Since minor arc  AX  = 35°,then angle AYX  =  35/2  = 17.5°

And since minor arc CY  = 80°, hen angle CXY  = 80 / 2  = 40°

Then  in triangle XBY, angle XBY = 180  -  17.5 - 40   =  122.5°

And angle  ABC  is vertical to angle  XBY....so its measure is also 122.5°

Second one

Since  in the larger circle, minor arc PQ  =  80  degrees

Then central angle POQ   also  = 80°

But if angle POQ  = 80°, then, in the smaller circle..... minor arc PQ  =  160°

Then arc POQ  =  360 -160   =  200°

First one

Connect RB

Then.....since   AR   = AB..... then   in triangle  ARB, angle ARB  = angle  ABR 

And  if  arc AB  = 130   then angle  ARB  = 130/2  =  65°   =  angle  ABR

So  angle  RAB  = angle PAB =   180 -130  =  50°

And since arc AQ = 28, then angle QBA  =  angle PBA  =  14°

So by the exterior angle theorem  angle  RPB  = angle PAB +angle PBA =  50 + 14  =  64°

a + b +  c  =  56       square both sides

a^2 + b^2 + c^2  + 2(ab + ac + bc)  =  3136      (1)

And

a^2  + b^2  + c^2  =  1344        (2)

Sub (2)  into (1)

1344  + 2(ab + ac + bc)  = 3136          subtract  1344 from both sides

2(ab + ac + bc)  =  1792             divide both sides by 2

(ab + ac + bc)  =  896            (a^2  =bc )

(ab + ac  + a^2)  = 896             factor out  a

a( a + b + c)  = 896

a ( 56)  =  896          divide both sides by 56

a = 16

And a^2  =  bc

So

abc  = a *a^2  =  a^3  =16^3  =   4096

sin(2x)  =  24/25

So

2sinxcosx =  24/25          square both sides

4sin^2(x)cos^2(x )  =  576/625

2sin^2(x)cos^2(x ) =  576/ [ 625 * 2 ]  =   288/625

cos(2x)   =   sqrt  ( 1 -  sin^2(2x) )   =  sqrt  ( 1 - 576/625)  =  sqrt ( 625 - 576] / 25  = sqrt (49) / 25 = 7/25

cos (2x)  =   cos^2x  - sin^2x

7/25  =  cos^2x - sin^2 x               square both sides

49/625  = cos^4x  -  2cos^2(x)sin^2(x) + sin^4x

49/625  =  sin^4x  +  cos^4x  -  288/625

[ 49  + 288 ] / 625   =  sin^4x + cos^4x

337/625  =  sin^4x + cos^4x

Ok I looked at the answer on my math program and it said to subtract 6 from both sides and then factor... so after subtracting 6 they got 
(x^2 + 2)(2y^3 - 3) = 143

and went from there.
It seems to work... 
huh

Huh ok, well thanks!

2x^2y^2 + 4y^3  =  149 + 3x^2      reaarange as

2x^2y^2  - 3x^2  + 4y^3  - 149 = 0        

WolframAlpha does not  show any integer solutions for x, y   :

https://www.wolframalpha.com/input/?i=2x%5E2y%5E2++-+3x%5E2++%2B+4y%5E3++-+149+%3D+0++++++++

rounding to the nearest integer means anything from 7.5 to 8.49 will be rounded to 8

rounding to the nearest 10th means anything from 8.45 to 8.54 will be rounded to 8.5

   Then according to the first rule above, rounding to the nearest integer will result in 9

So any number from 8.45 to 8.49 would cause this scenario...rounding to the nearest integer will result in '8'   

    if you FIRST round to the nearest 10th you will round to 8.5     then rounding to the nearest integer results in '9' ....

You're welcome.....good luck with your science studies! 

When x = 0    p(0) = 2

p(0+2)-p(0) =6

p(2)-2= 6

p(2)=8

p(2+2) -p(2)=6

p(4)=14

I think p(x) = 3x+2   ??

thanks! very helpful.

The atom is visualized as having spinning electrons around the nucleus of the atom.   The nucleus contains protons (the same number as the number of electrons unless the atom is an ION) . Protons determine the atomic number of the element.  The nucleus also contains a variable number of neutrons (thus creating isotopes).

Square both sides of the first equation:

(x²+y²)² = 1²    or   x⁴+ 2(xy)² + y⁴ = 1

So:  17/18 + 2(xy)² = 1 

       2(xy)² = 1/18

         (xy)² = 1/36

         xy = 1/6   or   xy = -1/6

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