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a^2 b^3 = 32/27     (1)      

a/b^3 =  27/ 4    ⇒    b^3 =  (4/27) a     (2)

Sub (2) into (1)  and we have that

a^2 (4/27) a =   32/27     simplify

a^3 (4/27) = 32/27

a^3 = ( 32/27) (27/4)

a^3 = 32/4 

a^3 = 8

a = 2     and   

b^3 = (4/27) (2)    =   8 /27

So....taking the cube root of both sides

b = 2/3

So

a + b  =  2 + 2/3  =  8 / 3

We can write the equation of this line as

y = -2 (x - 5)

y = - 2x + 10         to find the y intercept....set x = 0

y = -2(0) + 10         

y = 10

So...the y intercept is    (0, 10 ) 

We would have these  lines

y = 2x

x = 2y

Look at the graph here : https://www.desmos.com/calculator/szklgrm0br

There are 4 regions

Hi   IneedHALP, 

Thanks for interacting with me :)

There is one when they are both on the SAME side of the tangent.

and there is one when the are on OPPOSITE sides of the tangent.

draw IT !  AND you will see.

Do you know what all the words mean?  Like tangent and radius?

1)  Draw the tangent first. (That is just a line)

2)  Now put the point H on it.

You need two pics. When when the circles are both on the same side.

and on opposite sides

draw it and work out haw far apart the centres are.

Note: a tangen is a line that only just touches the circle. It only makes contact in one place.

Order shouldn't matter

For instance....in the first part

2 blue, 3 red  and 5 green  is the same as

3 red , 2 blue and 5 green,  is the same as

5 green, 3 red, 2 blue....etc,

I think that the order matters too, so wouldn't there be more? 

Here's my best attempt...

a)

Possible sums of 10 =

2  3  5               

2  4  4              

2  6  2   

3  3  4        4 possibilities

For each of these  there are   3 possible colors for the first card, 2 for the second and 1 for the third = 3 * 2 * 1 =

6 possibilities

So.....

4 possbilities * 6 possibilities = 24 ways    

b)

Possible sums of 20

9 9 2        8  8  4         7  7  6       

9 8 3        8  7  5          

9 7 4        8  6  6

9 6 5 

So   again....we have 8 possibilities * 6 possibilities =    48  ways     

The answer is

4* ceiling [ 78000/24000]  =     16 tires

oh so i should weigh them XD

if not for measuring the weight how else would i be able to know if it has evaporated?

No idea.

The water is going to be mixed in with everything, and absorbed by the sponge etc.

There's no visual measurement that will let know accurately know the water level.

Is weighing them a problem? 

The items are distinguishable, the bags are not

Enumeration

4 items is any bag  =  1 way

3 items in one bag, 1 in any other   = 4C3 =  4 ways

2 items in one bag , 2 in another = 4C2 / 2 = 3 ways

2 items in one bag , 1 in each of the other two = 4C2 = 6

14 ways

ahaha i agree!

Im a little bit slow in science please bear with me 

you previously mentioned 

"Since the gravel/sand/sponges don't change weight you are essentially measuring the evaporation rate."

if not for measuring the weight how else would i be able to know if it has evaporated?

An idea for a biology project might be to study the effect of increasing salinity on the development of freshwater organisms.

I was just reading that this is becoming a problem for North Carolina.

how would i be able to display my results since it will be in an enclosed room.

I don't know what you mean by this.  Do you mean how do you take weight measurements w/o disturbing the system?

You don't.  You just have to go in and take the measurement carefully and accept your interference as a source of uncertainty.

$50 won't get you robots to sit inside the room and do it. :D 

(8q^3r^2 + 6qr^2 + 22q) + (−6qr^2 − qr^2 − 14q − 4q^3r^2)

Combine like terms

( 8q^3r^2 - 4q^3r^2) + (6qr^2 - 6qr^2 - qr^2)  + ( 22q - 14q) =

4q^3r^2  - qr^2 + 8q

xxxxxxxxxxxx.

1.

Since AS is tangent to the circle  triangle AOS is a right triangle and  AO = 1....then OS =  sqrt [ (12/5)^2 + 1 ] = sqrt [ 144/25 + 1 ] =

sqrt [  144/25 + 25/25 ] =   sqrt [ 169 / 25 ] =  13/5

So....

OS - OP = PS

13/5 - 1 = PS

13/5 - 5/5 = PS

8 / 5  =  PS

i definitely would take your project idea into consideration. I was maybe leaning towards biology but it doesn't have to be that. 

also how would i be able to display my results since it will be in an enclosed room. Should i just show my data results?

I work with horses.  Our corrals get flooded during rain and at some point evaporation becomes the

main way they dry.  One of the things I have to do is spread the puddles from the low areas out so they'll dry faster.

The experiment should show that evaporation from the 100% sand is fastest, and evaporation from the 100% water is very slow.

This is why we use sand in corrals.

Drainage and evaporation also comes up in all construction, though it's more drainage there.  At least if done correctly.

I suppose there are more exciting projects.

When I judged these things I looked more for good experiment design and data analysis than for bells and whistles.

We'd get projects where the parents clearly funded and helped that were real fancy but the write up showed the kid

had no clue.  I didn't rate those well.  Whereas we'd also get shoestring projects with a clear hypothesis and a

well designed experiment and good write up that I would rate very well.

You have any areas that you'd like to pull your project from?  Any real world problems concern you?

In a 30- 60 - 90 right triangle, let the side opposite the 30° angle = 1

Then the side opposite the 90° will be twice as long = 2

Then the side opposite the 60°  angle =  sqrt (2^2 - 1^2 } = sqrt (3)

So  the tangent of 60° =  opposite side / adjacent side   =  sqt(3) / 1 = sqrt (3)

Thank you.

thank you very much! just one question..

how is  evaporation rates an  important real life problem.  i fail to understand that.