All Answers 352934 Answers

Lines    Regions

  1           2

  2           4

  3           7

  4          11

Using just the regions we have and the sum of differences we have

2      4        7            11

    2        3         4

         1         1

We have   2 non-zero rows ....so   we will have a second power polynomial in the form

an^2 + bn  + c      and we have these system of equations

a  +  b  +  c   = 2

4a  + 2b + c = 4

9a  + 3b + c  = 7

Subtract  the first equation from each of the other two and we have the system

3a + b = 2     ⇒  -6a - 2b  = -4    (a)

8a + 2b  = 5          (b)

Add (a) and (b)  and we have

2a  = 1

a = 1/2

And 

8(1/2) + 2b =5

4 + 2b =5

2b =1

b =1/2

And (1/2) + (1/2) + c = 2

 1 + c  = 2

c =1

So....the  resulting polynomial for n lines  is

(1/2)n^2 + (1/2)n  + 1  =

[ n^2 + n  + 2 ]

___________

        2

So....for 9 lines we have

9^2 + 9 + 2              92

_________  =       ____  =    46 regions

       2                      2

Do you follow the lines or do you go through the boxes? 

_?

excellent  

_.

No

Imagine the  focus is the Sun

The nearest that Mercury is to the Sun is 46 million miles....this is the distance  a - c

The farthest that Mercury is to the Sun is 70 million miles....this is the distance a + c

So  adding these distances we have  (a - c) + (a + c)  =  2a  =  the length of the major axis =116 million miles

... This man is a god. There is no denying it. You are simply a god.

The discriminant  is

12^2  - 4 (2) (23)  =

144 - 8*23  = -40     (as you found)

We need the discriminant to be  =  0   for the vertex to lie on the x axis

Note  that   in the form   ax^2 + bx + c...  only "c"  affects the translation vertically

Therefore we needt to  have

12^2 - 4(2) c  =  0

144 - 8c  =  0

144 = 8c       divide both sides by 8

c = 18

So...this means that  if we translate the function down by  (23 - 18)  = 5  units....it will have its vertex on the x axis

See the graph  here to confirm this : https://www.desmos.com/calculator/dtqz9ykt4k

Thanks Phil! I get it now lol :))

Let a = b = c.  Then the quadratic is az^2 + az + a = a(z^2 + z + 1) = 0.  The roots of z are -1/2 \pm sqrt(3)/2*i, and these have magnitude 1, so this is the maximum.

WOW! great job lol does it work?

You can do this in 2*3*4 = 24 ways.

There are three triples: (7,0,0), (0,7,0), (0,0,7).

This is what I get: x=5, y =3, 10*5+3=53: 3 x 5 x 53 =795

You are very welcome and I am glad I could be of help.

i have anothe rquestion, wouldn't the answer for the first one be 70-46?

Thank you

See here :

https://web2.0calc.com/questions/cphill-i-need-your-help-please

1)  What is the distance between the perihelion and the aphelion?

70 +  46   =   116  million miles

2) What is the distance from the center of Mercury’s elliptical orbit and the Sun?

The major axis  is 116 million miles

This  =  2a

So

2a  = 116

a  =  58

This is the distance from the center of the ellipse to one endpoint on the major axis

And  c  = the distance from the center of its elliptical orbit to the Sun   =  58 - 46  = 12 million miles 

3) Write the equation of the elliptical orbit of Mercury, where the major axis runs horizontally. Allow a and b to be measured in millions of miles. Use the origin as the center of the ellipse.
 

b  = sqrt [ 58^2 - 12^2]  =  sqrt (3220)

b^2  = 3220

a^2  = 58^2  = 3364

The equation is

x^2                y^2

____   +       ____  =      1

3364             3220

4)  Eccentricity   =  c/a =  12 /   58    ≈   .207

5)   The closer to 0 the eccentrcity is,   the more circular the ellipse....the nearer to 1, the more elongated

Thus.....the elliptical  path here is somewhat circular

Here's a graph :   https://www.desmos.com/calculator/cj4ey4e2gx

Homework Answers will cost $4.20, pay up!

.

That is the 420 pixel picture of my totally legit transaction ID for pay pal.

You can send your cash that way 

YES! That is what I was trying, proving that there are 30 60 90 triangles within the shape, then solving!

Since ABC is isosceles....Angle FAP   = Angle EAP  = 30°

EA = FA

PA  = PA

So....by SAS....triangle FAP = triangle  EAP 

And Angle EAP  = Angle ECR   =  30°

So angle CGA  = 120°

Draw DF   and angle  DQF  = 120°

And angle QDF  = angle QFD  = 30°

So triangles CGA and  DQF are similar

Draw GE

And EA  = sqrt (3)  / 2

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

So.....AG  = EA * 2 * sqrt (3)  =  sqrt (3) / 2 * 2 / sqrt (3)  = 1

And triangle BDF is similar to triangle BCA

And since BF = (1/2) BA, then DF = (1/2)CA

And since CGA and DQF  are similar....then  DF  = (1/2) CA  so   QF = (1/2) AG = 1/2

And AG  is the side of regular  hexagon DREPFQ

So.....the perimeter  is  6 (AG)  =  6 (1/2)   =  3

Thank you for tackling the Probability questions! 

Actually? Wow, AoPS uses Web2.0Calc as a source? Interesting!

I am from AoPS, but I dont take courses, I just self-grind. Its a pretty boring life, but its the same as working out.

Those body-builders grind teh gym to fatten their muscles, people like me grind AoPS to fatten my brain.