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thank you!!

The women can enter in   5!  =  120 ways

And the  men can enter   is 4!  = 24  ways

So    (120) (24)   =  2880  ways

u 2?

Hey, jugoslav.....could you show how you solved this  ???

I couldn't figure  out  any way to solve it  

Here are two answers  : https://web2.0calc.com/questions/a-researcher-observes-and-records-the-height-of-a

The first one  is exactly the same problem as yours

The second  is a little different.....the numbers  are just changed a  little

Let me know if you get stuck  .....

Cost  for 5 lbs =  

5.75 ( 5)  -  3.25  =  $25.50

To buy  p  pounds  of  coffee

C (p)   = 5.75 (p)  - 3.25

∠CED = 90º       ∠ADE = 67.5º          ∠ACE = 90 + 67.5 = 157.5º

∠BAO + ∠AEC = 180 - 157.5 = 22.5º

∠BAO : ∠AEC = 2 : 1

Angle BAO = 15 degrees 

Put the  0   in any  one of the trailing  four positions

And the  rest of the digits can be arranged in  4!/2!  identifiable ways  =   24/2 =  12  ways

So  4 * 12  =   48.....just as the guest found  !!!!

How many ordered pairs of positive integers (m,n) satisfy gcd(m,n) = 22 and lcm(m,n) = 1100.

[22 x 1100] = 24200 = 2^3 * 5^2 * 11^2 =[4 x 11] =44  and  11 x 25 x 2 ==550

Therefore:

m = 550  and  n= 44, or:

n = 44     and  m=550  

See here :

The  greatest  distance   is  just   BA + AD  =  BD  =  33 +  37   =  70

[(2, 0, 4, 5, 5), (2, 0, 5, 4, 5), (2, 0, 5, 5, 4), (2, 4, 0, 5, 5), (2, 4, 5, 0, 5), (2, 4, 5, 5, 0), (2, 5, 0, 4, 5), (2, 5, 0, 5, 4), (2, 5, 4, 0, 5), (2, 5, 4, 5, 0), (2, 5, 5, 0, 4), (2, 5, 5, 4, 0), (4, 0, 2, 5, 5), (4, 0, 5, 2, 5), (4, 0, 5, 5, 2), (4, 2, 0, 5, 5), (4, 2, 5, 0, 5), (4, 2, 5, 5, 0), (4, 5, 0, 2, 5), (4, 5, 0, 5, 2), (4, 5, 2, 0, 5), (4, 5, 2, 5, 0), (4, 5, 5, 0, 2), (4, 5, 5, 2, 0), (5, 0, 2, 4, 5), (5, 0, 2, 5, 4), (5, 0, 4, 2, 5), (5, 0, 4, 5, 2), (5, 0, 5, 2, 4), (5, 0, 5, 4, 2), (5, 2, 0, 4, 5), (5, 2, 0, 5, 4), (5, 2, 4, 0, 5), (5, 2, 4, 5, 0), (5, 2, 5, 0, 4), (5, 2, 5, 4, 0), (5, 4, 0, 2, 5), (5, 4, 0, 5, 2), (5, 4, 2, 0, 5), (5, 4, 2, 5, 0), (5, 4, 5, 0, 2), (5, 4, 5, 2, 0), (5, 5, 0, 2, 4), (5, 5, 0, 4, 2), (5, 5, 2, 0, 4), (5, 5, 2, 4, 0), (5, 5, 4, 0, 2), (5, 5, 4, 2, 0)] >Total distinct permutations = 48 ways.

I don't really have  a tuttorial on desmos

You  can download it from  their  site.....you can log in by supplying an email and password  so you  can share your graphs   (I believe these are "saved" so  it makes  logging in each time easy )

Just  type whatever equation , points, etc.,   you want  in the   boxes on the  left

If you  check  the label box  under each entry  it  will  display point coordinates.....you can also  customize this by typing  some identifier  on the  line  by the  label box   (for ex :   A = (1,0)  ).....you can add anything  on this  label line 

I like desmos  OK,  but I tend to use Geogebra  more.....it is  a little harder to  use, but  it is ideal  for  drawing segments, etc......you  might explore  it, as well

5n+14≤13

steps: 

subtract 14 from both sides and divide both sides by 5 to get n≤-1/5 (decimal form: n≤-0.2)

@Guest: 2A is A multiplied by 2

Thankyou Guest and CPhill,

CPhill could you actually share a link on how to use desmos.You have made it easy to understand/solve many geometry related questions, but I am not sure on how to use desmos to add the  points A and O in this case or plot the circle :)

Good job geoNewbie  !!!

See the graph here : https://www.desmos.com/calculator/6wrak7ngnw

Notice that a cricle centered at (8, -2)    with a radius of  sqrt (68)   will go through  both the  origin and  (10-10)

So  PA  = PO   =  sqrt (68)

I see no errors in your work, and your reasoning seems sound at first glance. I think you, geoNewbie21, have solved this problem correctly.

I like your use of symmetry to figure out the other x-intercept. I wish I had thought of this ingenuity...

BD = 33 + 37

 One x intercept is   x = 8

Because of symmetry, the  other x intercept  will  be   2 - (8 - 2) = 2  -6  = -4

And the point  (2, -54)  is on the graph so we  can solve  for "a"  thusly

-54 =  a ( 2 - - 4)  ( 2 - 8)

-54  = a  ( 6)(-6)

-54  = a ( -36)

a  =  54/36   =  3/2

Here's a graph   :  https://www.desmos.com/calculator/48496ve0xe

There may be quicker ways to do this question, but this is how I approached it.

The vertex form of a parabola is written \(f(x)=a(x-h)^2+k\)  where \((h,k)\) are the coordinates of the vertex of the parabola. The coordinates are given as (2, -54), so h = 2 and k = -54.

\(f(x) = a(x-2)^2-54\\ f(x)=a(x^2-4x+4)-54\\ f(x)=ax^2-4ax+4a-54\)

From the form \(f(x)=\text{__}(x-\text{__})(x-8)\), it is clear that (x-8) is a factor of \(f(x)\), which also implies that \(f(8)=0\).

\(f(8)=64a-32a+4a-54\\ 0=36a-54\\ a=\frac{3}{2} \)

Now we know the value for a, so we should be able to factor \(f(x)\) into a form that looks like \(f(x)=\text{__}(x-\text{__})(x-8)\) now.

\(f(x)=\frac{3}{2}x^2-4*\frac{3}{2}x+4*\frac{3}{2}-54\\ f(x)=\frac{3}{2}x^2-6x-48\\\)

To ease the factoring process, I will factor out 3/2 from all the terms.

\(f(x)=\frac{3}{2}(x^2-4x-32)\\\)

Now, factoring is significantly easier than before.

\(f(x)=\frac{3}{2}(x+4)(x-8)\)

From the factored form of the f(x), the x-intercepts are at (-4, 0) and (8, 0). Also, assuming that the value of a corresponds to the leading coefficient of this quadratic, then a = 3/2.

Good luck on your solving adventures.

its ok!

Unless I am seriously misinterpreting the question, this question is nonsensical. A is a vertex of the triangle, so 2A makes no sense.

The  distance from  D to T   represents    3/ (3 + 1)   = 3/4  of the  distance  from D to F....we  can  find  T  as

( 1  + (3/4)(7-1)   ,   4 + (3/4) (1  - 4)  )    =

( 1  + (3/4)(6)   ,   4  + (3/4) (-3)  )   =

( 1  +  18/4   ,    4  -  9/4 )   =

(22/4  ,  7/4)  =

( 11/2 , 7/4  )   =  T

Thank you so much, have a great life