CPhill

I just  made  a  typo in the original problem....I used  38  instead of 39.....here is the  corrected  answer  :

3)

If angle B = 39 degrees and arc PS = 116 degrees, find the degree measure of arc AS.

Angle B =  (1/2) ( arc AS - arc AP)

39  =  (1/2) (arc AS - arc AP)

78  =  arc AS - arc AP     (1)

And

arc AS + arc AP  + arc PS  = 360

arc AS + arc AP   +  116  =  360

arc AS  + arc AP  =  244   (2)

Add  (1)  and (2)

arc AS - arc AP   =  78

arc AS +  arc AP  = 244

2 arc AS  =  322      divide by 2

arc AS  =  161°

p =  27 - (x/400)^2  =  [ 27  -  x^2 / 160000]

(a) The revenue function for a week  is  price * pairs sold  =  p *  x   =   27x  - x^3/160000

(b)   The  domain is  [0, c]

And we can  find  c   as follows

  27x  - x^3/160000   =    0

x ( 27 - x^2 / 160000)   = 0

x = 0  is one solution   [ we  know this  ]

And

27  = x^2  / 160000   =  0

27   = x^2  /160000

(160000)(27)  =   x^2      take  the positive  root

x ≈  2078  = c

So  the domain is   [ 0,  2078 ] 

(c) Here is the  graph  : 

https://www.desmos.com/calculator/suhpihdav5

Let  the normal  rate  of  the  plane  = P

Let the rate of the wind  = W

With the wind, the effective  rate =  (P + W)

Against the wid, the effective rate  is (P  -W)

And  Rate * Time  =  Distance

So

(P + W) * 4   = 700

(P - W)*5  = 700            simplify

4P + 4W  = 700     

5P  - 5W   = 700

Multiply  the  first equation   through by  5   and the  second through  by  4

20 P  - 20W  = 3500

20P -   20W =  2800     add these

40P  =  6300     divide  both sides  by  40

P =   157.5  mph

And 

 4 (157.5) +  4W  =700    divide through by 4

157.5  +  W  = 175   subtract  157.5  from both sides

W =  17.5

The possible rational zeroes  are

±  [  all the  factors  of  6  ]  / [ all the factors of  3  ] =

Factors of  6  =   1,2,3,6

Factors of 3   = 1,3

So  we  have  these

± [ 1/1, 2/1 , 3/1, 6/1, 1/3, 2/3  ]  =

± [ 1/3, 2/3 , 1 , 2, 3, 6 ]

True

y =  3x^2  -30x  + 76)

Complete the square on x

y  = 3 ( x^2  -  10x  +  25)  +  76  -  3*25

y  = 3  (  x  - 5) ^2   +  76   -75

y  = 3(x - 5)^2  + 1

Here;s the graph  :   https://www.desmos.com/calculator/vamctntdeo

The  vertex  is   ( 5 , 1)

The axis of symmetry ⇒    x  = 5

This parabola turns upward......the  vertex lies above  the  x  axis....so  there  are  no x intercepts

The y  intercept  =    76 

This seems to indicate that  your answer to  the first one is  good  :

https://en.wikipedia.org/wiki/Rudolf_Clausius

(x  -1)^4   =

(x - 1) (x -1) ( x - 1)  ( x - 1)

The zero  =  1       its multiplicity  =  4

Here's  a  graph  for the  first  one  :

https://www.desmos.com/calculator/eijmu9tzm9

csc A  =  r    /   y

r  = sqrt  [ (-4)^2  + (-6)^2  ] =  sqrt  [ 16  + 36]  = sqrt  [52]   =  sqrt   [4 * 13]  = sqrt (4) * sqrt (13) =  2sqrt (13)

So

csc A  =  2 sqrt (13)  /  -6    =     -sqrt (13)  / 3

(a) How many different line segments are formed by joining pairs of of these points?

I don't know the answers to (b)  or (c)   but for this one we have

n ( n -1)        10  *  9

________  =  _______  =      45

    2                     2

See here  for a  simple  derivation of the  formula  :

https://www.basic-mathematics.com/number-of-segments.html