CPhill

5 hrs 20 min =  5 + 1/3 hrs =  (16/3)hrs

Call the rate of the plane in still air =  R

With the wind the rate is R + 75

Against the wnid the rate is R - 75 

Call the distance between the cities D

D/R = T

In the first instance

Total Distance / Rate in still air = 16/3 hrs

2D / R   =  (16/3)

D/R = 8/3

D = (8/3)R      (1)

In the second instance

Time from Penthaven to Jackson  + Time from Jackson to Penthaven  =  6  hrs

D / ( R + 75 )  +  D / ( R - 75)  =  6      (2)

Sub (1)  into (2)

(8/3)R / ( R + 75)  + (8/3)R / ( R -75)   = 6

(8/3)R ( R -75)  + (8/3)R (R + 75)  =  6 ( R+ 75)(R -75)

(8/3)R^2 - 200R + (8/3)R^2 + 200R  =  6 ( R^2 - 5625)

(16/3)R^2  = 6R^2 - 33750

R^2 ( 6 -16/3)  = 33750

(2/3)R^2 = 33750

R^2  = 50625

R=  225

The distance between the  cities 

D = (8/3)225 =   600  miles 

f (f ( f(x)))  =  66x - 47

f (f (x) ) =  p(px + q) + q =  p^2x + pq + q

f ( f ( f (x)) )  =  p (p^2x + pq + q) + q =   p^3x + p^2q + pq + q

So

p^3x + p^2q + pq + q  =  66x - 47

p^3  = 66

p= 66^(1/3)

And

p^2q + pq + q  =  -47

66^(2/3)q + 66^(1/3)*q + q = -47

q (66^(2/3) + 66^(1/3) + 1)  =  -47

q  = -47 / ( 66^(2/3) + 66^(1/3) + 1)

p + q =     [ 66^(1/3) ( 66^(2/3) + 66^(1/3) + 1)  - 47] / [ 66^(2/3) +66^(1/3) + 1 ]  =

[ 66^(2/3) + 66^(1/3) + 19 ]  /  [66^(2/3) + 66^(1/3) + 1 ]  ≈  1.842..... 

The x coordinate of the vertex =  -1  =    -b / (2a)

So

-b / ( 2 * 2)   = -1

-b / 4 =  -1

b/4 = 1

b  = 4

Simplify as

y = -5x^2 + 15x - 10

x coordinate of the  vertex  = (-15) / (2 * -5)  = 3/2

y coordinate of the  vertex  = 

 -5(3/2)^2  + 15(3/2) - 10  =  

-45/4 + 45/2 - 10 =

45/4 -10  =

5/4

Vertex = ( 3/2, 5/4)  = ( 1.5 1.25)

\( f(x) + b \cdot g(x)\)

If b = 0, the resulting polynomial will be degree 5

There cannot be more than 3 yelllow squares

Four ways

Y B Y R Y

Y R Y B Y

Y B Y B Y

Y R Y R Y

      1     =     row 0

   1    1

 1   2    1

1  3    3    1

The sum of the entries in the nth row of Pascal's Triangle  =  2^n

The number of entries in any row  =  n + 1     

So  the average is

[ 2^n ] / [ n + 1]  = 2

2^n  = 2( n + 1)

2^n =  2n + 2

Note that  this is  true if n = 3

Angle AED = 90  = Angle BED

DC = DE

DB  = DB

Triangle DBC congrunent to triangle DBE  by HL

tan (DBC)  = DC / BC  = (BC/3) / BC   = 1/3  = tan (DBE)

tan (EBC)  =  AC  / BC  = 

tan (2 * DBC)   = 

 [ tan (DBE) + tan (DBC) ]  / [ 1 - tan(DBE)*tan (DBC) ] =

(1/3 + 1/3 )  /  ( 1 - (1/3)(1/3) )  =

(2/3) / ( 8/9)  =

18/24 =

3/4 =  AC / BC

We have a lower-power polynomial divided by a higher-power polynomial.....in such a case the rational function will approach 0 as x approaches infinity, but there will be no  definitive minimum

(x - 5) (x + 5)  =  x^2  + 0x  +  -25