CPhill

4%   = 4/100 =   .04

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. 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 )

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"

We  have this function :

P(t)  =  100(e)^kt

And when t  = 5, we have that

1000 =  100(e)^k5            divide both sides by  100

10  =  e^5k        take the ln of both sides

ln 10  =  ln e^5k     and we can write

ln 10  =  5k  ln e                  {  ln e   =  1...so we can ignore this }

ln 10 =  5k        divide both sides by 5

ln 10 / 5  =  k

The function is

P(t)  = 100(e) ^{ln10 * t / 5}

So.....we want to find the doubling time....and we have that

200  = 100(e)^{ln10 * t / 5}       divide both sides by  100

2  =  e ^{ln 10 * t / 5}      take the ln  again

ln 2  = ln e ^{ln 10 *  / 5}      amd we can write

ln 2  =  [  (ln10) / 5 ] * t          

5 ln 2  =  ln 10 * t

5 ln 2 / ln 10   =  t  ≈     1.5 weeks

What Melody said  !!!!!

(-5)(-3)-(2)(3)

--------------------

3[5/(3-8]

Let's take the top, first

(-5)(-3)  - (2)(3)  =

15  -  6    = 

9       

The bottom is a little trickier

3 [ 5 / (3 - 8) ]  =         work inside the parentheses, first

3 [ 5 / (-5) ]       =       now, inside the brackets  

3 [  -1 ]    =           now, just multiply

-3

So.....we end up with

9  /  - 3    =

-3

-8 + 4- (-23) -13

Rememer that   a  "-"  outside any parentheses or brackets changes the sign(s)  on the term(s) inside....so we have.....

-8  +  4  + 23  -  13  =

- 4  + 23 - 13    =

-4  +  10

6