All Answers 358181 Answers

Graphically I have found the answer to be about 8

Here is my interactive graph.  You can move the point P around to get different areas.

A=4k+1

A=7m+1

A=5t+2

7 is the biggest one so I just looked at the 7 ones and see if they worked for the others.

8,15,22,29,36,43,50,57  Yep, 57 works for the others as well,  so that is the smallest one

Let the balls be Red Blue and Yellow.  My method requires that the ball be thought of as different

the pairs are   RB, RY and BY  so there  are 3 ways to pair them.

Number of ways the balls can go into the cups is 10*10*10 = 1000

Number of ways to get Blue and Red together and Yellow somewhere else = 10*1*9=90

there are 3 possible pairings    so that is 90*3=270

So P(exactly 2 in the same cup)  = 270/1000 = 27/100

Another way to look at it is by subtraction:

the number of ways they can all be in different cups is 10*9*8 = 720

The number of ways they can all be in the same cup is 10*1*1 =10

so the number of ways that 2 will be in a cup is 1000-720-10 = 270

the answer is the same so that is conclusive enough for me.

For specific positive numbers m and n, the quadratics 16x^2 + 44x + 56 and what?? differ only in their constant term. What is mn?

Do the expansion and equate coefficients.

Draw a grid 12 by 12 (plus an extra row and an extra column for the 2 numbers rolled).

Fill in the squares and see how many add to more than 16   etc

P(www)+P(RRR)

P(WWW)=  \(\frac{4}{9}*\frac{3}{8}*\frac{2}{7}\)

You can continue from there

Triangle ADC  is a Pythagorean  Triple  6 - 8 -10   right triangle ....so AD  = 6

And  in right triangle BDA ,   hypotenuse BA  = sqrt (6^2  + 12^2)  =  sqrt ( 180) =  6sqrt (5)

So....sin B =   AD / BA  =     6 / (6sqrt (5))  =   1 /sqrt (5)  =  sqrt (5)  / 5 

Two chords, AB and CD meet inside a circle at P If AP = 3 and CP = 7 then what is BP/DP?

BP/DP = 7/3

Use Law of Sines to find  angle C

sin C  / 8 =  sin150 / (7sqrt (3))

sin C  = 8 sin (150)  / (7sqrt (3))

sin C = 8 (1/2)  / (7sqrt (3))

sin C =  4  /  (7sqrt (3) )

arcsin C =  [ 4 / (7sqrt (3)) ]  ≈  19.3°

Angle B=   180  - 150  - 19.3   ≈  10.7°

Using the Law of sines once more

AB /sin B =  BC  / sin A

AB / sin (10.7) =  7sqrt (3) / sin (150)

AB =   sin (10.7) * 7sqrt (3)  / (1/2)

AB =  sin (10.7) * 14 sqrt (3) ≈  4.5

1/ cosx * sin x

______________________

sin x / cos x  + cosx / sinx

sin x/ cos x

________________

sin^2 x         cos^2x

________ +_________

sinxcosx       sinxcos x

sin x/cos x

____________

sin^2 x + cos^2x

______________

sin x cos x

sinx / cos x

___________

         1

  _________

   sin x cos x

sinx / cos x   *   sin x cos x

We can use  the intersecting chord theorem here

AP * PB = CP * PD

3 * PB = 7 * PD

PB/ DP  =  7/3

  CA  = b =  side opposite angle B

 CB  = a = side opposite angle A

Using the Law of Sines

sin B / b =  sin A / a

sin B / (4sqrt (2) )  = sin A  / (4sqrt (3))

sin B  /  (4 sqrt (2))  =  sin 45° / (4sqrt (3))

sin B  / (4sqrt (2))  = (1/sqrt (2) )  / (4sqrt (3))

sin B =  (4sqrt (2) ) / (4sqrt (3))  * (1 /sqrt (2) )

sin B   =    1/ sqrt (3)

arcsin (1/sqrt (3))  = B ≈   35.26°

-7pi/3   +  2pi  =                                       4pi/7  + 2pi

-7pi/3  +  6pi/3  =                                     4pi/7  + 14pi/7  =

-pi/3                                                          18pi/7

Flip 'em to base 10     log 3/ log 4 = x

log 27 /log8 = kx = k log3/log4

k = log27/log8 * log4/log3 = 2

log ₄ 3  = x      log ₈ 27  = kx

Put  these in exponential form

4^x  = 3          8^kx  = 27

(2²)^x =  3       (2³)^kx  = 27

(2^2x)  = 3      (2^2x) ^1.5k = 27

                      (3)^1.5k  = 27

                      (3)^1.5k = 3³

Solve for the exponents

1.5k =  3

k  = 2

k =2

Do you mean the sum of the middle row?

3+2n-9+n

= +2n+n   +3-9

=     3n         -6

Now the top row was   2n + 1

and the middle row is   3n - 6

so

2n + 1 = 3n - 6

2n + 1 -2n= 3n - 6 -2n

           + 1 = 1n - 6

     + 1 +6 = 1n - 6 +6

           7   =  n

            n = 7

Oops... sorry, I meant to say that the minor arc must have 360-340 =20 degrees of measure, so it is 1/18 * 2 * 18 * pi = 2pi

Somehow I found the wrong area using the diameter ^^

12 players must play 16 games (two against 8 others) + 3 games (their own squad)= 19 games

12 x 19 = 228   BUT   TWO players are in each game ...so divide by two = 114 games of chess are played

There are two distinct solutions x to the equation 18 + 5x^2 = 30x. If each solution is rounded to the nearest integer, and then these two integers are multiplied together, what is the result?

Hello Guest!

\(18 + 5x^2 = 30x\\ 5x^2-30x+18=0 \)

a          b           c

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

\(x = {30 \pm \sqrt{900-4\cdot 5\cdot 18} \over 2\cdot 5}\\ x=\frac{30\pm \sqrt{540}}{10}\\ x_1=5.32379\approx5\\ x_2=0.67621\approx 1\)

\(x_1\cdot x_2\approx5\cdot 1\\ \color{blue}x_1\cdot x_2\approx5\)

  !

-pi/3

and

18pi/7

I'm gonna correct your question first:

Your question: Let A be an acute angle such that sin 4A = sin 5A. What is the smallest possible measure of A in degrees?

Corrected question: Let A be an acute angle such that sin 4A = cos 5A. What is the smallest possible measure of A in degrees?

Angle A = 90 / (4 + 5)        ==>       ∠A = 10º

Proof:

sin 4A = sin 40º 

                                     sin 40º = cos 50º

cos 5A = cos 50º

If I understand your question. 2 12-sided dice are rolled, what is the probability of getting a sum > 16?

The probability is [8 + 7 + 6 + 5 + 4 + 3 + 2 + 1] / 12^2 =36 / 12^2 = 1 / 4