CPhill

Note that the total possible arrangements are  5!  =  120

Let's  look at the arrangements where AD do sit together

AD - - -

-AD - -

- - AD -

- - - AD

So  they could occupy  4 different positions....and for each of these they could be seated in 2 ways (AD or DA)

 And for each of these.....the other 3 can be seated in 3! ways

So.....the number of seatingss where they don't  sit together  =

120  - (4)(2)(3!)  =

120   - 8*6  =

120  - 48  =

72

Sudsw....from what you describe, see the image :

PC = 3

*******

We have the form 

P  =  ab^t         where  a = the original population =  27    and t is the number of days

So....we have

36  = 27b^3      divide both sides  by  27

(4/3)  = b^3      take the  3rd root

(4/3)^(1/3)  = b     

So   we want to find this

P  =  27[ *4/3)^(1/3) ] ^30

P =  27 (4/3)^10  =  exact value

P   ≈  479  bees after 30 days

We can find the radius  of the  circle, R, by using the Law of Cosines

We have that

(9 sqrt (3))^2 =  2R^2  - 2R^2 ( cos 120°)

243  =  2R^2 - 2R^2 * (- 1/2)

243 =  2R^2 + R^2

243  =  3R^2 

Divide both sides by 3

81  = R^2   

Then the area of the  circle =

pi R^2   =

pi *81  =

81 pi  units^2

Q1

4log₃ c + 4log₃ d  =   4 [ log₃ c + log₃ d ]    =   4 log₃ (cd) ]   = log₃ (cd)⁴    TRUE

(3/4) [ Ln a  + Ln b ]  =  (3/4) Ln (ab)  =  Ln(ab)^(3/4)  =  Ln   ( ⁴ √  [a³b³ ] )   FALSE

3 Ln e  + 2 Ln f  =   Ln e³ + Ln f²  =   Ln  (e³ f² )    FALSE

Q2

Note that the points  (0, 1) and (3, 6.4)    are  on the original function

So....the points  (1, 0)  and (6.4,, 3)  are on the one of the  graphs shown

This has to be the graph on the  bottom left

This solution can be easily  found  graphically : https://www.desmos.com/calculator/8tv2ssgvrv

The area = two triangles each  with a base of  4  and a height of 2

So.....the area  =   2  [ (1/2) (4) (2) ]  =   8 units^2

See the graph  here :    https://www.desmos.com/calculator/qeidqnvtts

The intersection of this line and the circle  will form the diameter of the  circle....to find the radius we can  complete the square  on x  in the equation of the circle......so we have....

x^2 - 12x  +36  + y^2   = 28 + 36

(x + 6)^2  + y^2 =  64

The center is  (6, 0)

The radius  = sqrt (64)   = 8

So....the area we are looking for  is

(1/2) pi (8)^2 =   

32 pi units^2

Subbing the expression  for x  into the radical, we have that

x = √[  x^2 +  √[2020 - x ] ]      square both sides

x^2  = x^2 + √[2020 - x ]

0  =  √[ [2020 - x ]

x = 2020

We have these integers

4, 8,  9,  12, 16, 18, 20, 24, 25,  27, 28, 32, 36, 40, 44, 45, 48, 49, 50

(19  integers )