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If  the  shaded  regions  are equal.....then  the  chords  that comprise  the  top and  sides  of  the  trapezoid  must  be  equal

Call the  top two  vertices A and B   and  the center  0

Then . connecting  the top  two  vertices  to  the center   will  form  triangle  AOB  with  angle  AOB =  60°   and  OA, OB  =  1 m

Using  the  Law  of Cosines  we  can find   chord AB   as  :

AB^2   =  OA^2  + OB^2   -  2(OA * OB) cos (AOB)

AB^2  =  1^2  +  1^2   -  2 ( 1 * 1) (1/2)

AB^2   =  2 - 1

AB^2  =  1

AB =  1

The   perimeter of  the  trapezoid  =  3AB   +  diameter   =   3 (1)  +  2   =    5 m

5000/131 = 38 and a bit more

N = 131x + 11 

x can be from 0 to 38, 39 answers. :))

=^._.^=

I did the  same  thing, EP

"p"    (in my case, N)   turned out to  be   460 /  3    ???

Cue  "Crazy Train"  by  Ozzy Osburne   !!!!

The ratio of the number of marbles Kaden, Adam, and John gave away was 2:4:9. The number of marbles Kaden gave away was 2 fewer than the number of marbles he had left. Adam had 1/2 as many marbles left as John. Kaden had 1/2 as many marbles left as Adam. They had a total of 56 marbles left. How many marbles did John give away?

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

John gave away 27 marbles!

All marbles that are given away    6 + 12 + 27 = 45

All marbles left     8 + 16 + 32 = 56

3*arc BC   +  arc AD   =   360°           ( 1)

(1/2) (arc BC  -  arc AD)   =  50°        (2)

angle  ACD   = (1/2) arc AD              (3)

Manipulating (2)   we  have  that

arc BC  -  arc  AD   =   100°

arc  BC  =  100°  + arc AD          (4)

Sub  ( 4)   into   (1)

3 ( 100  + arc AD)  +  arc AD  =   360°

300  +  4 * arc AD   =  360° 

4 * arc AD   =   360  -  300

4arc AD  =   60 °

arc AD   =   60 / 4   =   15°

Using (3)

angle ACD   = (1/2) (15°)   =   7.5°

it is

from -2 to 5   is a distance of 7     we want 5/7 of this to get to 3   (from 2)

   for a   from 7 to -8 is a 'distance' of -15         5/7 * (-15) added to 7 = -75/7  + 7 =  - 26/7      ( as Chris found ....!)

2, -8    is not on the line    y = 2x/3

                 it IS on the parallel line    y =  2x/3 - 9 1/3         multiply both sides by 3

                                                    3y = 2x - 28

                                                 -2x + 3y = -28             Is that what you need?

p = original count of people

1/4 got off leaving   3/4 p

  5 people got on   making it  3/4 p + 5

     then 1/2 got off     leaving  1/2      1/2 (3/4 p + 5)

        and 201 got on for a total of 80     1/2 (3/4p +5) + 20 = 80     Now all you have to do is solve this for 'p'

                                                                                                   can you do that ?

PA  =  PB  because   tangents  drawn  from an  external point  to  a  circle  are equal

Triangle   PAO  is  right with  angle PAO  = 90

So   PO   is  the  hypotenuse,     leg PA   = 15     and  leg  OA  =  the  radius

So

PO  = sqrt  ( PA^2   +  OA^2)   =  sqrt  ( 15^2  +  9^2)   =  sqrt  306   =  3sqrt 34

The  area  of  this  triangle   =    (15 * 9)  / 2    =    135/2

And    PBOA   will form a kite  with an area of  twice  [ PAO ]   =  135

And  PO  and  AB   will   be  the  diagonals  of  this kite

And  the  area   of  this kite  =   product of diagonal lengths  / 2

So

135   =   3sqrt 34  * AB  /  2

270   =  3sqrt 34 * AB

AB =   270  /  (3sqrt 34 )  =   90 / sqrt 34    units   ≈   15.43 units

14   different  products

To see this.....note  that  we  are  omitting   one  of  the  numbers  each  time

So.....the products   are

14!  / 1  ,    14!  / 2   ,   14! / 3   ........  14! / 12  ,    14!/ 13   ,   14! / 14   =    14 different  products

Wait... how come it is 2 years? I thought it said 12 months.

You're right, jugoslav; that wording is strange! 

sure i know it must be something like (a-b)(c-d) where a, b, c, d, are some radical numbers, at least one of them must have a sqrt2, and at least one must have a sqrt3

Wouldn't it be 22/36?

The  slope  of  the line  is

[ -8 - 7 ]  /   [ 5 - -2 ]    =   -15 / 7

Since   ( 3, a)  is  on  the line....we  can solve fro "a"   thusly

[ 7 - a ]  /  [ -2 - 3 ]     =    -15/7

[ 7 - a ]  /  - 5   =  -15/7              multiply through  by   -5

7 - a   =  75/7         rearrange  as

7  - 757  =  a

(49/7) - (75/7)  =  a

-26/7   =  a

We  first  need to  find  the  slope of L1

5x  +  8y   =  -9

8y =  -5x   - 9

y  = ( -5/8)x - 9/8

The slope  =  -5/8

L2  will  have  a negative reciprocal  slope =   8/5

Since L2  passes through (10, -10)  we  have

y  = (8/5) ( x - 10)   - 10

y =  (8/5)x  -  16  -   10

y = (8/5)x  -  26

m   +  b   =      8/5  -  26   =  -122/5

OOOMMMGGGGG TYSM CPHILL!!!!!!! :D

Here's  what I get  :

In  one  tournament   we  have

C(7,3) *  [ C (7, 2) ] ^3   =

35 * 21^3   =

324135   possible committees in one tourney

I will assume they both invested 1200   and the '4' in your question was supposed to be a dollar sign

After 16 years

Liam has      1200 (1 + .0875/12)^(16*12)

Camilla has  1200 e^16(.0921)                                You can take from here I expect......

There is probably a smart way of doing this, but I'm just going to list the 27 "words."

1. AAA

2. AAB

3. AAC

4. ABA

5. ABB

6. ABC

7. ACA

8. ACB

9. ACC

10. BAA

11. BAB

So the answer is 11.

I guess the "smart" way of doing this is recognizing that numbers 1-9 start with A, 10-18 start with B, and 19-27 start with C. BAB is the 2nd word that starts with B, so 9+2=11.

Nice problem!

The first card could be anything and it doesn't matter. However, the second card has to be of the same rank. There are 4 cards in every rank, so there would be 3 left. There are 52 cards in the deck, but after the first card is chosen, there are 51 left. Therefore, the answer is 3/51 = 1/17.

There are 10 options for the President. 
 

For the Vice President, there are 9 options (anyone but the President)

For the Secretary, there are 8 options (anyone but the President and Vice President).

This gives us: 10 * 9 * 8 = 720.

If a person could only have at most 1 office at a time, then we would be "choosing" 3 people from the 10. This is why tmatthews answered \(\binom{10}{3}\). However, since a person can have more than one office, then there are 10 options for the President, 10 for the vice President, and 10 for the secretary. This gets us 10 x 10 x 10 = 1000.

Find all the values of x such that \(\frac{4x-5}{3x+5} \geq 3\).

Keep in mind that an equality can be solved like an equation. The only difference is that whenever we multiply or divide by a negative number, we have to switch the greater than or equal to sign to a less than or equal to sign.

We multiply both sides by 3x+5 to get rid of the denominator and get: \(4x-5 \geq 3(3x+5)\).

We then expand out the right hand side: \(4x-5 \geq 9x+15\).

We subtract and add terms to get all the constants (numbers) on one side and all the variables on the other: \(-20 \geq 5x\).
Finally, we divide by 5 to get x by itself: \(-4 \geq x\).