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Thank you. :))

I just enrolled to the amc 12 class with a friend, it may be slightly too hard for me, but I'm willing to study a bit more. 

I've been on AoPS for a while (just not the classes), and I really love the company and I feel like the class will be amazing too. 

=^._.^=

z-score   of -1    is   one standard eviation from the mean and = .1841     or     18.41 %

                                       18.41 are below    and 100-18.41 =81.59 % of the scors are higher

My best attempt  :

P(ace)  =  4/52  = 1/13

P(heart  but  not the ace of hearts) =  12/52  =  3/13

P(any other  card)  = (52 - 4 - 12) / 52 =    36/52 =  9/13 

Expectation  =

(1/13)(2.70)  + ( 3/13) (1.40)  - (9/13)(.10)    ≈  .462

Sorry CPhill, but your answer is wrong.

thank you

For f(1) = 0, it has to contain (x-1). 

That leaves us with A and B.

For the vertex to be at the maxumum, the start number must be a negative, in this case, -3/2. 

So our answer is B.

I hope this helped. :))

=^._.^=

distinct permutaions = 

Number of letters  !   /   [  (Repeats ) !  ]   =

7!   /   [   ( repeats of A)!  ( repeats of R)!     ]  =

7!  / [ 2!  * 2!   ]  = 

5040  /  4  =

1260  distinct permutations

Notice  that it appears that every month, the total cost increases  by  $30

And  the  starting cost  = 80 =   $50 fixed  cost + $30 variable cost

So....letting C   be  the cost in any month and M  be  the  month, we  have  that

C =  50  + 30M

When M  =4, the total cost , C   =   50 + 30 (4)  =  $( 50 + 120)  = $170

For 1st month = 80

For 2nd month = 80 + 30

                         =110

For 3rd month = 110 + 30

                        = 140

For 4th month = 140 +30

                       = 170

So, for 4th month is $170

"Anchor"  a  member  of  either  party in one seat....for convenience, choose  a  Democrat to occupy  this seat....we have 5 choices

Going counter-clockwise, we  have 5 ways of choosing a member of the opposite party to sit  to the right of  the first person, 4  ways to choose  the person to sit to the  right of  the  second person, and so on

So.....the  total ways  =

D * R   D   R   D   R    D    R    D   R

 5*  5 * 4 * 4 * 3 * 3  *  2 *  2  *  1 * 1 =   25  * 16  * 9  * 4   =  14400  ways

Total no. of marbles = 6 + 3 + 4 + 1

                                = 14

a) P(3 red) = \({6 \over 14} * {5 \over 13} * {4 \over 12}\)

                  = 0.06

b) P(2 green) = 0 

c) P(blue then yellow) = \({6 \over 14}*{4 \over 13}\)

                                    = 0.13

d) P(1 red, 1 blue) = \(({6 \over 14}*{3 \over 13}) + ({3 \over 14}*{6 \over 13})\)

                              = \({36 \over 182}\)

                              = 0.2

q1 /

The area of the region under the graph of f(x)=4 + 1x over the closed interval 0≤ x ≤ 1 by choosing representative point  as the right endpoint of each subinterval is given by  

You  don't specify  the width of  the sub-interval, but we  can  choose a convenient value, say, .25

The  area  in this case is the sum of [ the  rectangle widths  *  their heights ].....because  we are using the right endpoints, the area  will be over-stated

.25 (4 + .25)   +  .25 ( 4 + .50) + .25   +  .25 ( 4 + .75)  + .25 ( 4 + 1)  =

.25  [  4.25  + 4.5   + 4.75 + 5 ]    =  4.625

BTW....the TRUE  area  is  given by    ( 4 ( 1)  +  (1/2) )  =  4.5

If we  made  the  widths of  thes sub-intervals smaller, we would get  a better approximation   ....

(1/2) abc    (1/2) cdf  =

(1/2 c)  ( ab - df)

7x^2 + 27xy - 4y^2  =

(7x - 1y) ( x + 4y)

Which  IS NOT  equal to  7x  + 27  - 4y

If 4 + i  is a root, so is  4-i

The polynomial will be of  degree 3  and will be  of  the form   ax^3 +bx^2  + cx + d

a  =   1

b  = the  sum of the roots  = - ( 3 + 4+i + 4 - i)  =  -11

c =  (3(4+i)  + 3(4 -i)  + (4 + i) (4 - i) )   =  ( 12 + 3i + 12 - 3i  + 16 - i^2)  =  (40 - - 1)   = 41

d =  -  [ (3) (4 + i) ( 4- i) ] =  - [ 3 ( 16 - i^2)]  = - [3 * 17]   =  -51

So....the polynomial  is

x^3  - 11x^2  +  41x   -  51

Thanks for posting back.  I'm the one who reproduced the question in purple, and suggested using a triangular pyramid as a model.  I didn't believe your subsequent explanation so I looked it up   You are correct.  I was mistaking edges for vertices.  A vertex of a triangle is one of the sharp points.

Hey there

The new price of the television will be 70% of its original. Thus

670*0.7=469

Hope this helped :)

2 2 1 is 15 ways not 10.

Total 41

Hey there!

I hope you see this as my answer was wrong!

A pyramid with 14 edges is also known as a heptagonal pyramid, meaning the base has 7 edges. Due to this, the 7 edges at the bottom and the 7 leading up to the point accumulate to 8 total vetices! Sorry for the confusion!

Hope this finds you well :)

Hey Guest!

Vertices are the point in which edges meet. So for your example of a triangular based pyramid, there would be 3 vertices at the bottom, where the edges meet, and one final vertice at the point, or tip if oyu will. Coming to a total of 4 vertices for a triangular based pyramid.

Hope this helped :)

P.S. while writing this, i realise I've actually given the wrong answer to this question. Edit time!!

Hey there!

If Daniel gets 4 out of 5 right thats a 4:5 ratio. If we let x be the number of questions he gets asked, and y be the number of questions he get right (64), we can solve for x.

\(x*\frac{4}{5}=y\)

\(x*\frac{4}{5}=64\)

\(x=\frac{64}{\frac{4}{5}}\)

\(x=80\)

Now you can figure out how many he missed!

Hope this helped :)

Hey there!

Here, the Surface Area (SA) will be equal to the base + the three faces. We know the formula for the area of a triangle (\(\frac{Base*Height}{2}\)), so we get:

\(SA=6.9 + 3(\frac{4*7}{2})=...cm^2\)

Putting this into your calculator will give you your answer.

Let me know how you go :)

\((2x+3)^2-(2x+5)(2x-5)=34\)

\((4x^2+12x+9)-(4x^2-25)=34\)

\(12x+34=34\)

\(12x=0\)

\(x=0\)

BO = 10          BC = 15

∠ACB = 2(arctan(10/15))

AB = tan(∠ACB) * 15

V = 1/3((BC)²pi * AB)