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Thanks very much Heureka :)

What a cute puppy !!

I don’t think it says division gives a polynomial Melody! There is a full stop (period) between the function definition and the word Give (which also starts with a capital letter), so I interpreted the sentence beginning with Give as an instruction to the reader, not a continuation of the previous sentence.

Thank you very much, Melody.

haha good

Hi Melody,

Your question is, "I will admit that I do not understand the second half of what you have done though Heureka. 

How did that combinatory maths sneak in there?"

1. The formula is from Jacob Bernoulli. Sorry the original paper from Jacob Bernoulli, Ars conjectandi ( Wahrscheinlichkeitsrechnung) page 100 is in german.

The frequency  is    1 / period

We have 3 periods in 2pi.....so.....the period is  2pi / 3

So....the frequency is

1 / [ 2pi /3 ]  =

3 / [ 2pi ]

If we allow for repeats....we have

20 consonants to choose from to fill the first position

6 vowels to choose from for the second position

20 consonants to choose from  for the third position

10 digits to choose from for the last position

So....the number of possible plates is

20 * 6 * 20 * 10   =

24000

1st one  :

This is the graph of the cosine reflected across the x axis....and with an amplitude of 1/2

So......the correct answer is :

f(x)  =  -(1/2)cos (x)

2nd one :

This is the sine graph with a period half as long as normal.....thus....there are 2 periods in 2pi

So....the correct answer is

f(x)  =  sin (2x)

Oh, ok. hahah

Thanks again

4%   = 4/100 =   .04

I just have one more question, where did you get the .04?

Thank you so so much!

I struggle with this and most of the times I don't know where to start.

Thank you very much! 

Total alcohol in the martini by mass

142 g x .30 = 43 g

15% of this goes into the bloodstream

42 g x 15% = 43g x .15= 6 gm    (if they drink the whole martini)

2. A hospital organized a blood donation event where 80 pints of blood were collected. The hospital expected to get 8 gallons of blood with the event. Has the goal been met?
1pt = 473mL
1 gal = 3785 mL

80 pints  =    80 * 473 ml  =  37840 ml

8 gal  =      8 * 3785   = 30280 ml

The goal has been met  !!!!

1. On the label of a bottle of vinegar, the information appears: "acetic acid solution at 4% by weight". If the bottle contains 750mL of vinegar and its density is 1.006g / mL, calculate the mass of acetic acid contained in the bottle.

Density  =  Mass / Volume    .....     which means that  .....

Volume * Density  =  Mass

So....for the liquid in the bottle we have that

750 ml  *  1.006 g / ml     =   Mass         [  the "ml"  cancels ]

750 *  1.006 g  = 

754.5 g   ....   this is the mass ["weight "] of the liquid .....

And the acetic acid is 4%  of this

So........the  mass  ["weight"]  of the acetic acid  is 4% of this...so

.04 * 754.5  =

30.18 g

1.  a trapezoid

1. find the midpoint of the segment connecting the points (a,b) to (5a,-7b). (Hint: answer will contain variables a and b.)

Mid-point formula  :   [  add x coordinates / 2 , add y coordinates / 2  ]

So we have

[   ( a + 5a) / 2, ( b - 7b) / 2 ]  =

[ (6a)/2, (-6b)/2]  =

[  3a,  -3b ]

2..Find the coordinates of point P which divides the directed line segment AB with coordinates A(5,-2) and B(9,6) in the ratio 3:1.

A ratio of 3:1  means that we are dividing the distance between these two points into 4 equal parts

We are looking for a point P that is    3/4  of the distance from  A  and B

So  we have

[  5  +  ( 9 - 5)(3/4)  ,  -2 +  ( 6 - - 2) (3/4) ]  =

[  5  + (4)(3/4) ,    -2  +  (8)(3/4)  ]  =

[ 5 + 3,   -2  + 6 ]  =

( 8, 4 )

3.  Point C is on the directed line segment from points A(-11,4) to point B(-1,-6) and divides the segment in the ratio 2 to 3. Find the coordinates of C.

Like the last one, a ratio of 2 : 3  divides the distance  between these points into 5 equal parts

and we are looking for  point C that is 2/5 the distance from A to B

So   we have

[  -11 +  (-1 - -11)(2/5) ,  4 + ( -6 - 4) (2/5)  ]  =

[  -11 +  (10)(2/5) ,  4 + (-10)(2/5) ] =

[ -11 + 4 , 4 + -4 ]  =

( - 7, 0 )

Ok, Thank You!

1.

midpoint of  (a, b)  and  (5a, -7b)   =   ( [a + 5a]/2 ,  [b + -7b]/2 )   =   ( 6a/2 ,  -6b/2 )   =   ( 3a, -3b )

We have a right triangle with an opposite side of 96......the hypotenuse is the distance we want

So....the sine function is what we need

So we have

sin (6°)  =  96 / D       rearrange as

D  =   96 / sin (6°)  ≈  918.4 ft

I think this is correct...but I'd like to have someone else check my answer  [ and make corrections, if necessary ]

Looking at it from the other poiint of view, we are asking...."What is the fewest number of dice  I need to roll so that the chances of not rolling a "6"  are  ≤ 10% ??"

The chances of not rolling a "6"  with one die  = 5/6

So.....the chances of not rolling a "6"  with two die  =  (5/6)^n

So.....we need to solve this :

(5/6)^n  ≤ .10

We can make this an equality

(5/6)^n  =  .10

Unfortunately.....I don't see a way to avoid using logs  !!!  [though there may be one ]

Take the log of both sides

log (/5/6)^n  =  log .10      and we can write

n log (5/6)  =  log .10       divide both sides by log (5/6)

n  =  log (.10)  / log (5/6)  =  12. 62  ≈ 13

So.... we need   13  dice to have a 90% probability of rolling at least one "6"

So far I have gotten  (5/6)^x = 0.1, using complementary counting. I used a Calculator to solve for x and I got 13 as the final answers since you can't roll a die a decimal times.