CPhill

focus = ( 2,2)

directix = y =  -4

The directrix is below the  fosus so this parabola will turn upward

The form we are  looking  for is

4p ( y - k)  = (x - h)^2

(h , k) is the  vertex   .....this is  given  by  ( x coordinate of focus , ( y coordinate of focus + directrix) / 2)  =

(2 , ( 2 +-4) / 2)  =  ( 2 ,-2/2)  =   (2, -1)

p =  the distance  between the fcus and the  vertex = y coordinate of focus  -  y  coordinate  of  vertex  =

2 - -1  =   3

So....the equation is

4(3) ( y - - 1) = ( x -2)^2   →    12 (y + 1) = ( x - 2)^2

Here's the graph  :  https://www.desmos.com/calculator/xik9fowbq0

y = x^2  - 2

Let   the   point on the parabola  be  ( x , x^2 - 2)

Using the distance formula  we  have  that

D =   [ x^2  +  ( x^2 - 2)^2  ]^(1/2)       simplify

D = [ x^2  + x^4  - 4x^2 + 4] ^(1/2)

D =  [ x^4 - 3x^2 + 4 ) ^(1/2)       take the  derivative of this  and set to 0

D'  = [ 4x^3 + 6x ] / [ x^4 - 3x^2 + 4 ] ^(1/2)  = 0

So

4x^3  + 6x   =  0

2x (2x^2 - 3)  = 0

2x  = 0                                   2x^2 - 3  = 0

x  = 0   reject                          2x^2  = 3

                                                 x^2 = 3/2

                                                 x  = ±sqrt (3/2)

When x =  sqrt (3/2)    then y  =  [sqrt (3/2)] ^2 - 2 =  -1/2

When x = -sqrt (3/2)  y has the same value = -1/2

So....the shortest distance in either case is

sqrt  ( 3/2  + 1/2)  = sqrt (2)

See the graph, here : https://www.desmos.com/calculator/i36wydpg70

Arc length =   r  θ

6  = r  ( 3pi/5)

6 (5 / (3pi) )  =  r   =  10/pi  ≈  3.18 m

Area of parallelogram  = DC *  EG =  27 * 16   =  432

Also  area =  FH * AD =  FH * BC

FH * BC   =432

FH * 18  = 432

FH = 432  / 18    =  24

A_sector = pi * r^2 *  ( 115 /360)  = pi * 17^2  * (115 /360)    ≈   290 ft^2

We can write

(sqrt (3) )^2  =  3

(sqrt (4)^3  =  (2)^3  = 8

(sqrt (5)^4  = (sqrt (5))^2 * ( sqrt (5))^2  = 5 * 5   =25

So we  have

3 * 8 * 25  =     600

Simple interest  amount after 4 years at 6% =  20000  +  20000 (.06) (4)   = $24800

Compound interst amount  after 4 years compounded quarterly at 10%  = 20000 (1 +.10/4) ^(4 * 4)  ≈  $29690 

[ 29690  - 24800 ]  = $4890

Very nice, Eternity    !!!!

(22 - 2)  =  20  take art alone

(19 - 2)  =  17 take band alone

                   2  take both

                   6  take  neither

                ___

                 45

2^(2x - 5)  =  18      

2^(2x)/ 2^5  = 18

(4)^x  = 2^5 * 18

4^x  = 576       lake the log of log of  both sides

log 4^x  = log 576

x * log 4  = log 576

x = log 576 / log 4

2^x =  2^ ( log 576 / log 4)  =   24