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 √[8a^3b^5 ]  =    √[ 4 * 2 * a^2 * a *  b^4 * b ]  =

(4)^(1/2) * (2)^(1/2) * (a^2)^(1/2) * (a)^(1/2) * (b^4)^(1/2) * (b)^(1/2) =

2 * (2)^(1/2) * a * (a)^(1/2) * b^2 * b^(1/2) =

2ab^2   √ [ 2ab]

 √[50x^3y^2 ] =   √ [ 25 * 2 * x^2 * x * y^2 ]  =

(25)^(1/2) * (2)^(1/2) * (x^2)^(1/2) * (x)^(1/2) * (y^2)^(1/2) =

5 * (2)^(1/2) * x * (x)^(1/2) * y  =

5 xy  √[ 2x]

AB + CD = 210   ⇒  CD = 210 - AB

Both triangles ACD and ABC are under the same height  ...so....

Area of triangle ACD                 3

_________________  =      ______

Area of triangle  ABC                7

(1/2) h (CD)              3

__________  =       ____

(1/2) h ( AB)              7

CD                  3

___    =          __

AB                  7

210 - AB                3

_______   =        ____     cross-multiply

   AB                       7

7( 210 - AB )  =   3(AB)       simplify 

1470  - 7AB  =  3AB       add  7AB to both sides

1470  =  10AB                divide both sides by  10

147  = AB 

Look at the graph here, GM : https://www.desmos.com/calculator/g2185zbyzy

Note that 2500 bacteria are reached at about 11.024 hrs  = 11 hrs (rounded)

BTW.....as you become more familiar with this....it won't seem that daunting ....it's not as difficult as it first appears !!!

For continuous growth..... we have this

A = A₀*e^(r*t)

Where A is the final number, A₀ is the original number, e = 2.718...., r =  hourly growth rate as a decimal, t = time in hours

So we have

2500 =  4 * e^(.584 t)    divide both sides by 4

625 =  e^(.584   t)    take the natural log, Ln, of both sides

Ln 625  = Ln ^(.584 t)        and by a log property, we can write

Ln 625  = .584t * Ln e       [ Ln e  = 1....so....we can ignore this ]

Ln 625  = .584 t       divide both sides by .584

Ln 625 / .584  = t ≈11.02 hrs = 11 hrs

First one.....convert to exponential form....

(50r^2s^4)^(1/2)  =   ( 25 * 2 * r^2 * s^4)  =  (25)^(1/2) * (2)^(1/2) * (r^2)^(1/2) * (s^4)^(1/2) =

5 * (2)^(1/2) * r * s^2  =

5 r s^2 √2

Second one      3 √[18x^3]  =  3  √ [ 9 * 2 * x^3 ]  =   3 *(9)^(1/2) * (2)^(1/2) * (x^2)^(1/2) * (x)^(1/2)  =

3 * 3 * 2^(1/2) * x * (x)^(1/2)  =

9 * x * 2^(1/2) * (x)^(1/2)  =

9x  √ [ 2x ]

Done!!!!

Your answer of 68  is still good    !!!

Could you edit your answer? Sorry just to double check

Sorry....I mis-read that....it should be 16%

Can you take it from there or do you want me to edit my answer to check???/

Correct  !!!!

5x - 7x^2 = 3x + 4    rearrange as

7x^2   - 2x +  4   =    0

Using the quadratic formula  with a = 7, b = -2  and c = 4     we have that

x  =  - (-2) ±  √[ (-2)^2 - 4*7 *4]             2 ±  √[ 4 - 112]            2 ±  √[-108]

        ______________________  =   ______________ =   ____________   =

                   2 (7)                                          14                             14

2 ± i √[ 9 * 3 * 4 ]             2 ±  6i√3             1 ± 3i √3

______________  =     _________  =     _________

           14                            14                         7

where did you get 18% isnt it 16%?

Let x be one of the equal sides  and y be the remaining side

We have this system

2x > y

2x + y = 20

Look at the graph :  https://www.desmos.com/calculator/rmoylu7ibl

It shows that there are 4 such triangles

6, 6, 8

7, 7, 6

8, 8, 4

9, 9, 2

Is that jsut for the other question ?

Since your online again, I was kind of confused on this one as to why you said " Since this isnt your choice it is 68"

Ok I fixed it, sorry about that. ):

No its okay you have been plenty of help maybe ElectricPavlov can help me on it when he gets online. Im gonna repost the question in case he doesnt see it here. Thank you again

OK.....glad to help.....sorry I don't know the answer(s) to the first one....I hate to "guess" and maybe give incorrect answers......

I see it now! That was a confusing but from your explaintions it makes more sense. Thank you for your help CPhill