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The blue graph is just f(x) shifted up 3 units....so...its equation is...

y = f(x) + 3

It looks pretty symmetrical.....    the negative values will cancel the positive values        = 0

CU...it might be easier if we  just considered this :

{BTW....this is not a "square " ...it's a rhombus]

Note that triangles ABC  and FGC are similar

And the height of ABC  =  sqrt (3)

And note that FG   is 1 unit above AB

So....the height of FGC  must just be  [ sqrt (3) - 1]

So

height of FGC         1/2 base of FGC

___________  =    _______________

height of ABC           1/2 base of ABC

[sqrt(3) - 1 ]          (1/2 ) base FGC

__________  =     _____________

    sqrt (3)                         1

So  (1/2) base   FGC  =     [ sqrt (3) - 1] / sqrt (3)

So...its area must be   its height * (1/2) its base  =

[ sqrt (3) - 1 ] [sqrt (3) - 1 ] / sqrt (3)  =  [ 3 - 2sqrt (3) + 1 ] / sqrt (3)  =  [ 4 - 2 sqrt (3)] / sqrt (3)

So...by symmetry.....the area of the rhombus must be twice this  = [ 8 - 4sqrt(3)] / sqrt (3)  units^2

Yahhhh... I still can't really 'fold' my brain around it ! 

If the side length is 2...the center of the square is   1,1     ....is that what you are looking for????

Nice, EP.....that's a tough one to visualize   !!!!

Six sided die only has digits  1  2  3  4  5  6   on it.

No....the integers on the dice cube range from 1 to 6

We will have an 8 digit number.....but no digit > 6

Thanks, EP.....I hadn't thought about that method   !!!!

Never done one like this before.....but....I believe that the rotation of this will produce two cones

The  volume of an octahedron is given by

V  = √2/3  * edge length^3

243* 3 / √2  =  edge length^3

729/√2  = edge length ^3

9 / (2)^(1/6)  = edge length 

The distance from any vertex to its non-adjacent vertex  = √2 edge  = 9√2 / {2)^(1/6)  = 9∛2

The radius of each cone will be (1/2) of this  =  4.5∛2

And this will also be the height of each cone

So....the volume of the 3D figure will be  the volume of the two cones =

2 [ (1/3) pi * radius^2 * height ]  =

(2pi)  ( 4.5∛2)^2 (4.5∛2) / 3  =

2pi  [ 4.5]^3 * 2 / 3  =

(4pi] * [91.125] / 3  =

4pi (243/8)  =

(243/ 2) pi units^3

Now I see three is the answer.......when folded as a solid...I think this is the path to take:

So then how do you do it?

Thank you!

Wait, won't some of them contain digits larger than 6?

Sorry...... I didn't read the question very carefully !

When x = 0  (year 0 )   720500 represents the starting population

1.022 is the growth factor   1+.022  means   2.2 % growth/year

1548800 = 720500 1.022^x

log(2.1496) = log 1.022   * x

x = 35.17 years

1980 + 35.17 = 2015

Similarly.....treat   2,1 as a vector    slope = rise/run =   1/2       orthoganal slope is -2

-3,6     rise / run =   6/-3 = -2    orthogonal

1,2     2/1 = 2    nope

-2,-3     -3/-2     nope

1, -2     -2/1     orthogonal

Yes, I am familiar with dot products... thank you! 

a=102;b=1;if(a%b==1, goto3,goto4);printa,b;a=a+10;b++;if( b<10, goto2, goto7);a=a+10;b=1;if(a<10000, goto2, discard=0;

I found 6 such numbers:

162 7, 322 3, 622 3, 862 7, 922 3, 982 9 - The first number is the "dividend" and the smaller number is the "divisor".
 

We have orthogonal vectors when their dot products  = 0

So

(2, 1) dot (-3, 6)  =  -6 +6  =  0    orthogonal

(2, 1) dot (  1, 2 )  = 2 + 2   = 4    not  orthogonal

(2, 1) dot ( -2, - 3)  = -4 + -3  =  -7     not orthogonal

(2, 1) dot ( 1, - 2)  = 2 + -2  =  0       orthogonal

[ I'm assuming that your are familiar with dot products ....]

Here's your answer with the Shoelace Theorem

(-3, 2)       (6 , - 2)        (3, 5)

 (x1, y1)    (x2, y2)      (x3, y3)

The area is given by :

abs [ x1y2 + x2y3 + x3y1  - x1y3 - x2y1 - x3y2 ]

_________________________________________   =

                              2

abs [ (-3*-2) + (6*5) + (3*2) - (-3^5) - (6*2) - (3 * -2) ]

_____________________________________________   =

                             2

abs [ 6  + 30 + 6 - (-15)  - 12 - (-6) ]

_______________________________

                       2

abs [ 63 - 12 ] / 2  

51 / 2  =

25.5 units^2

Thanks for turning me on to this  !!!!.....I want to look at it a little more...I believe that it just involves a vector cross-product ......

This is a NET!

You are both taking the long route   !

4 nPr 1 x 4 nPr 3 x 3 =

     4      x     24    x 3 = 288