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The difference between 2 numbers in this set will always be even because an odd number minus an odd number will always be even.

The smallest attainable value is 2, and the biggest attainable value is 18. Every value between these 2 numbers will also be attainable.

That means the answer to this question is: there are 9 distinct positive integers that can be represented as the difference of 2 numbers in that set.

(they are 2, 4, 6, 8, 10, 12, 14, 16, and 18)

The triangle on the left is a 30-60-90 right triangle

Its height =   8 * sqrt (3)/2   =  4sqrt (3)   =  sqrt (48)

The triangle on the right is a 45-45-90  right triangle.....the altitude and  base of this triangle are equal  since both are opposite 45° angles

So....by the Pythagorean Theroem

x  =  sqrt  [ (sqrt (48) )^2  +  (sqrt (48))^2   ]  =   sqrt  [2 (48)^2 ]  =      48sqrt (2)

4) Find the remainder when f(x) = 6x^3 - 29x + 12 is divided by x-6

            6x^2     + 36x    +   187 

x - 6  [  6x^3   +  0 x^2    -   29x    +   12   ]

            6x^3    - 36x^2

           __________________

                        36x^2   -  29x 

                        36x^2 -  216x

                      __________________

                                    187 x  +  12

                                    187 x  -1122

                                   _____________

                                              1134  = remainder

1) Find the remainder when g(x) = 5x^9 - 4x^2 +2 is divided by 5x-10.

                      x^8   + 2x^7   + 4x^6 +  8x^5 + 16x^4 + 32x^3 + 64x^2  +  ( 636/5)x + 1272/5

5x   -10    [   5x^9   + 0 x^8  + 0x^7 + 0x^6 + 0x^5 + 0x^4 +  0x^3  - 4x^2  + 0x +  2   ]

                    5x^9  -10x^8

                  _____________________

                               10x^8  +  0x^7

                               10x^8   -  20x^7

                              ____________________

                                            20x^7  + 0x^6

                                            20x^7  - 40x^6

                                         __________________

                                                        40x^6  - 0x^5

                                                        40x^6  - 80x^5

                                                       __________________

                                                                     80x^5  + 0x^4

                                                                     80x^5  - 160x^4

                                                                    __________________

                                                                                160x^4 + 0 x^3

                                                                                160x^4  - 320x^3

                                                                               _________________

                                                                                               320x^3 - 4x^2

                                                                                               320x^3  - 640x^2

                                                                                              _________________

                                                                                                              636x^2 + 0x

                                                                                                              636x^2  - 1272x

                                                                                                             ___________________

                                                                                                                            1272x   +  2

                                                                                                                             1272x  - 2544

                                                                                                                           ______________

                                                                                                                                         2546  =  remainder

Thanks!

The answer to (3)  is here  :

https://web2.0calc.com/questions/math-help_61676

Keep in mind that we only need 1 example to disprove something.

A. As one angle determines all the other ones in a parallelogram, A is true, because all the other angles are 180-90 = 90, so it is a rectangle.

B. True, same reason as E

C. True, I think same reason as E

D. True, opposite angles are equal, so all the angles are auto-completed, making the equal-sided rhombus a square.

E.  True, as the diagonals must bisect each other (definition of a parallelogram) and so all the small triangles formed by the diagonals are congruent by SSS and have largest angle 360/4=90˚. Because a parallelogram's angle bisectors are diagonals, and all the triangles are isosceles right, each angle is 45*2 = 90, and as at least two sides of a parallelogram are equal, and each angle is 90 from part A, we have that all the sides and angles are equal, so it is a square.

Connect  OC

Then   triangle BOC is isoceles  with   OC   =   OB,,  so angles OBC  and  OCB are equal

And central angle COB   =180  -2(70)   =  40°

So angle BAC  is an inscribed angle  intersecting the same arc as angle COB...so it measures (1/2) 40  = 20°

Note  that  arc   BCD  = 2 (40)  = 80°

And  angle BFD  =  (1/2)arc BCD   =  40°

And angle CED  =  angle BAC  =  20°

So.....BAC + BCD  + CED   =   20  +  40   +  20   =   80°

A point lies above \(y=2x+7\) if its y -coordinate is greater than 2 times its x-coordinate plus 7. Checking the given points, we find that (6,20), (12,3), and (20,50) satisfy this condition. The sum of the x-coordinates of these points is \(6+12+20=\boxed{38}\). 

a) f ( x + 7)....this one shifts  the  vertex to the left by 7 units....so  the  vertex is  ( 2 - 7, -3)  =  (-5,-3)

b) f (-x).....this reflects  the  vertex across the  x axis  =   (-2,-3)

Slight math error on my part....-2.5  = -5/2 is correct...!!!

For convenience, let the  bottom  vertex of the  square  =(0,0)

The slope of one of the  intersecting lines is 2...and  its equation is y  =2x

The slope of anoteher of the intersecting lines  is   -1/2.....and its equation is   

y = -1/2x  +  2

The x  intersection of these  lines  is

2x =  -1/2x  +  2

(5/2)x   = 2

x =4/5   =   .8

And the  y coordiante is     t = (2)(4/5)   = 8/5  =  1.6

We  have a triangle with  the points   ( .8, 1.6)  , (1, 2)   and (2,2)

The (squared) sides of this triangle  are

 [ (.8- 1)^2  + ( 1.6 - 2)^2  ]  = .2

[ (.8 - 2)^2  + (1.6 - 2)^2 ]  = 1.6

And 1

We can use the Law of Cosines to find the angle

1 =  .2  + 1.6  - 2  ( sqrt ( .2) sqrt ( 1.6) cos  (angle)

1 =  .2  + 1.6  - 2  sqrt ( .32) cos (angle)

[ 1 - ,2  -1.6 ] / [- 2sqrt (.32) ]  = cos (angle)

arccos   ( [ 1 - .2 - 1.6 ]  / [- 2sqrt (.32) ] )   =   45°    = angle

thank you so much i got the first part and worked out the function but got -2.5 but thank you

The point (0,-14)  is on the graph

Thus in the form x^2 + bx + c.     c  =  -14

Also, the point (2,0)  is on the graph  which means that we have

0 = (2)^2  + b(2)   -14      simplify

0 = 4 + 2b   - 14

0 = -10 + 2b

10  = 2b

5   = b

So  the  function is   ax^2  + bx + c    =   1x^2  +  5b  -14 

The  turning point occurs at the vertex......the  x coordinate is given by   -b  /(2a)  =  -5/ (2 * 1)  =  -5/2   

Hello Guest!

Another way of thinking about drawing 3 balls, is not drawing 1 ball. Since there are 4 balls and 3 colours, we can represent them as A, A, B, C. 

To draw 3 different balls, we just need to not draw an A ball. Since there are 2 A balls, the chance of not drawing an A ball is 1/2. 

I hope this helped. :)))

=^._.^=

Isolate y in 3x + 4y = -14 ==> y = 3x/4 - 14/4

Slope of Perpendicular = -4/3, point-slope form of equation y - 7 = -4/3(x + 5)

==> 3y - 21 = -4(x + 5) = -4x + 20

==> 3y = -4x + 41

==> y = -4/3*x + 41/3

m + b = -4/3 + 41/3 = 37/3

You are more likely to get a good answer if you present your question in a readable format.

\(T=11\sqrt{M}\\ \frac{T}{11}=\sqrt{M}\)

Now square both sides.

The second one is easy.  The last one is not so bad either.

I assume you are being asked to make x the subject.

I'll start half way through your working.

\(y - 3x = 50x^2 - 60x + 26\\ y = 50x^2 - 57x + 26\\ 2y-52 = 100x^2 - 114x \\ \frac{2y-52}{100} = x^2 - 1.14x \\ \text{Complete the square}\\ \frac{2y-52}{100} +(0.57)^2= x^2 - 1.14x+(0.57)^2 \\ \frac{2y-52}{100} +0.3249= (x-0.57)^2 \\ \frac{2y-52+32.49}{100} = (x-0.57)^2 \\ \frac{2y-19.51}{100} = (x-0.57)^2 \\ \pm\frac{\sqrt{2y-19.51}}{10} = (x-0.57) \\ x=0.57\pm\frac{\sqrt{2y-19.51}}{10} \\ x=\frac{5.7\pm\sqrt{2y-19.51}}{10} \\\)

You will need to check for careless mistakes.

Convert these to base 10.

the base 10 equation is    A=6B-22

it was base 6 so A and B must both be 0,1,2,3,4,or5

so solve this.

\(0\le6B-22\le5\)

there is only one possible answer.

Then solve for A

             ABA

          + 411

   =      1215

we must have A + 1 = 5, so A = 4

Hence

          4B4

     +   411

  =    1215

or 4*6² + B*6 + 4 + 4*6² +  1*6 + 1 = 1*6³ + 2*6² + 1*6 + 5 

I'll leave you to calculate B from this.

OR alternatively ....

rationalize the denominator first by multiplying top and bottom by sqrt5

\(\sqrt{\frac{\frac{5}{\sqrt{80}}*\sqrt5+\frac{\sqrt{845}}{9}*\sqrt{5}+\sqrt{45}*\sqrt5}{\sqrt5*\sqrt5}}\\ \sqrt{\frac{\frac{5}{\sqrt{16}}+\frac{\sqrt{5*169}}{9}*\sqrt{5}+\sqrt{5*9}*\sqrt5}{5}}\\\)

etc

Ignore the biggest square root till the end

Simplify each term on the numerator separately like this.

\(\frac{5}{\sqrt{80}}=\frac{5}{\sqrt{16*5}}=\frac{5}{4\sqrt{5}}\)

etc

Now you need to rationalize the denominator so multiply every term, top and bottom by   sqrt5

Keep going from there.   It all falls out quite nicely.