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9 a^8 x^10 - 6 a^8 x^9 + a^8 x^8 + 72 a^7 x^9 - 48 a^7 x^8 + 8 a^7 x^7 + 252 a^6 x^8 - 168 a^6 x^7 + 28 a^6 x^6 + 504 a^5 x^7 - 336 a^5 x^6 + 56 a^5 x^5 + 630 a^4 x^6 - 420 a^4 x^5 + 70 a^4 x^4 + 504 a^3 x^5 - 336 a^3 x^4 + 56 a^3 x^3 + 252 a^2 x^4 - 168 a^2 x^3 + 28 a^2 x^2 + 72 a x^3 - 48 a x^2 + 8 a x + 9 x^2 - 6 x + 1
(27 terms)

"Coin A is flipped three times and coin B is flipped four times. What is the probability that the number of heads obtained from flipping the two fair coins is the same?"

I get the following:

Edit:  Item number 3 in the list above should be a₂/a₁ not a₂/a₃ !

Good spotting guest.

I found my error, I did not read the question properly.    

P( at least 2 make it into the bin) 

= P(3)+P(4)+P(5)+P(6)

=1- [P(0)+P(1)]

Using the probabilities that I found before:

P(0)=\((\frac{2}{3})^6=\frac{64}{729}\)

\(P(1)=\frac{64}{263}=\frac{192}{729}\)

\(1-[P(1)+P(2)]=1-\frac{64+192}{729}\\ 1-[P(1)+P(2)]=1-\frac{256}{729}\\ 1-[P(1)+P(2)]=\frac{729-256}{729}\\ 1-[P(1)+P(2)]=\frac{473}{729}\\ \)

Now our answers agree.

Coding:

1-[P(1)+P(2)]=1-\frac{64+192}{729}\\
1-[P(1)+P(2)]=1-\frac{256}{729}\\
1-[P(1)+P(2)]=\frac{729-256}{729}\\
1-[P(1)+P(2)]=\frac{473}{729}\\
 

We can use the tangent  here

tan (32°)  = fountain height /  12      multiply both sides by 12

12 tan (32°)  = fountain height  ≈  7.5 ft

The  vector  can be described  as  S (90 - 41)° W =  S 49°W.  That is, a 49° angle in the 3rd quadrant using the (negative) y axis  as a a reference line .  So....we travel South from the origin and 49° W.

Look  at the graph here :  https://www.desmos.com/calculator/gd9gbzpdwn

The equation will  be true  for  10  lattice points  [points with integer coordinates ]  in the 3rd quadrant

And it  will  also  be  true  for   11^2  = 121  lattice points  on  the grid  

0  ≤ x  ≤ 10  and   0  ≤ y  ≤ 10   

So.....the total lattice points =  10  + 121  =   131

Use compass directions to describe the direction of the Vector. The diagram is not drawn to scale.  (Please note that this is a essay questions and shown work is needed.

Call the angle A

Note  that   tan A  =  opposite side / adjacent side  =   10/14

So...to  find A  take the tan inverse   (arctan)

arctan (10/14)   ≈  35.53°  =   36°

ok, another new one. :)   (make sure you fully understand the last one first though)

y=  0.5sin(2x) -1

1) midline   dashed line

2) topline (my term, not formal)           very light line    Hint:  it is a line, the equation needs any equal sign.

3) bottom line (my term, not formal)     very light line

4) wavelength (show a little working)

5) A quarter of the wavelength (this will help with choosing the scale on the x-axis)

NOTE: At this point you should have a rough sketch started

6) What would be sensible points for the x-axis? 

       Easiest to see this from your rough sketch that you are already making.

7) Will it start on the midline, above or under, why? 

                      So far you have just done this from memory of curve shape which is great. 

                       But, but you can just substitute.

                      when x=0    y=-2cos(0)+3 = ?

8) Will is start going down or going up, why?

9) y intercept

10) plot 4 (or 5) points in advance just by knowing what the properties of the graph will be.

                  (I mean maxima, minima, and points on the midline)

11) Graph it from x=0 to x= wavelength

12) what is the first maxima? (read from graph)

13) what is the first minima?

14) What points are on the midline?

Copy all these questions and answer on your own post. 

You can just copy my questions and answer after them if you want. (like last time)

You can check your sketch and answers with Desmos.  Make the x axis step equal to 1/4 of the wavelength.

Let n be the number of months when they will have the same amount

So  we have

145  +  10n  = 20  + 15n        subtract  10n , 20  from both sides

125  =  5n           divide both sides by 5

25 =  n

They will have the same amount  after 25 months

The total  number  of dominoes in this set    =  (18 + 1) (18 + 2)  / 2  =  19 * 20 / 2  =  190

There will 19  dominoes  that  are "doubles"  [  0,0    1,1   ......  18,18  ] 

So....the probability of selecting a double   is  just

19 / 190   =   

1 / 10

8 x 8 x 1  =   64  ft^3

6 x 4 x 1  =   24 ft^3

So....we have this proportion

48 / 64 =  x / 24        where x is the number of bags we need

3 / 4  =  x  /24             multiply   both sides by 24

18  =  x

We need 18 bags

Connect AD

Then triangle ADB  is right  with AB = 2 ,  AD  =1  and   BD  = √3

And in  triangle AEB, angle  AEB  = 120° 

And using the law of sines

EB / sin 30  = AB /sin 120

EB /(1/2)  = 2 / (√3/2)

2EB  = 4/√3

EB  = 2/√3  =  (2/3)√3

So   DB  - EB  =  DE  =  √3 - (2/3)√3  =  √3/3

And by symmetry DE = CE

And angle DEC  =120°

So the area of triangle  DEC =  (1/2) DE^2 * sin (120°)  =  (1/2) (√3/3)^2 * (√3/2)  =  √3/12    (1)

Next....let F  be  the  midpoint  of AB  

Connect   DF and CF

Angle DFC  =  60°

DF = CF  = 1

So the area  of sector DFC  = (1/2) (DF) (CF) * pi/3  = (1/2) (1)^2 * (pi/3)  = pi/6    (2)

And the area  of triangle DFC  = (1/2)(1) sin (60°)  = √3/4    (3)

So....the green  area =   (1) + [ (2) - (3) ]  =

√3/12  +  [ pi/6  - √3/4 ]   = 

pi/6  + √3/12  - 3√3/12  =

2pi /12 + √3 /12  - 3√3/12  =

[ 2pi   - 2√3 ]             pi  - √3

__________  =     _________ ≈   .235  units^2

      12                          6

a_{2009} is equal to 3 - 2*sqrt(7).

(a,b) = (3,-2).

haha

this one ez

i'm guessing you don't want a solution

ans is A,B,D,C

and it is false

See the following graph  :   https://www.desmos.com/calculator/gh92tovzqd

Let the number of children's tickets sold  = x

Let the number of adult tickets sold  = y

Five possibilities  are

(x , y )  =  

(0, 350)

(30, 320)

(52, 298)

(75, 275)

(300, 50)

A rectangular building is 4 meters by 6 meters. A dog is tied to a rope that is 10 meters long, and the other end is tied to the midpoint of one of the long sides of the building. Find the total area of the region that the dog can reach (not including the inside of the building), in square meters.

Total area is  248.1858196 m²  

Np!!! Thank you for the explanation of how the rational root thereom is used :D. Everyone makes mistakes. I do as well. I just keep practicing. <3 

P (orange on first draw  and  purple  on second)  =  (8/12 ) * (4/11)  = (2/3) * ( 4/11)  = 8 / 33

Here's my best attempt

We  have   C (20, 3)   = 1140  possible different sets of 3 integers

(a)  the sum will be even when   we have

even + even + even        or

even + odd +  odd

In the first case  we want to choose  any  3  of the 10  even numbers  =  C (10, 3)  = 120

In the second  case we want to  choose  any 1 of 10 evens  = C (10, 1)  and  any 2 of the 10 odds = C(10,2) = 45 

So  the probability  is

[  C (10,3)  +  C(10,1) * C(10,2) ]               120  + 10 * 45            570

____________________________  =      _____________  =   ______  =   .5

         C (20, 3)                                                1140                      1140

(b)  the only way we can have an odd product is when all three  integers are odd.....there  are C (10,3)  = 120 sets of these

So.....the probability that we have an even product =

1   -   120  / 1140  =

[ 1140  -120 ]  / 1140  =

1020 / 1140  = 

17  /19

No prob....glad to help   !!!!