CPhill

2. 

Area of Sector  = pi *r^2 *  (100  / 360)

250 pi  =  pi * r^2 * (5/18)

r^2  = 250 (18 / 5)

r^2  =  50 * 18

r^2  = 100 * 9

r^2 = 900

r  = 30

1.

Let O be the center of the semi-circle

Draw CE  perp  to AB

Connect OC  = r

OE = 8.5

EB = r - 8.5

CB = 3

Using the Pythagorean Theorem

Height of triangle OCE   = Height of triangle ECB

CE^2  =  CE^2

r^2  - (OE)^2  = (CB)^2  - ( EB)^2

r^2 - (8.5)^2  =  (3)^2 - (r - 8.5)^2

r^2  - 72.25  =  9 - r^2 + 17r - 72.25

r^2 - 72.25  =  -r^2 + 17r - 63.25

2r^2 - 17r -9  = 0

(2r + 1) ( r -  9)  =  0

r =  9 

(2 -n)(2-n) + (2n -4) (n + 1)  =  (n + 1) ( 2 - n)

n^2 -4n + 4  + 2n^2 - 2n -4  = -n^2 + n + 2

3n^2 - 6n  = -n^2 + n + 2

4n^2 -7n   - 2  = 0

(4n + 1) ( n - 2)  = 0

Set each factor to 0 and  solve

n = -1/4     and  n =  2  (reject this answer....it makes a denominayor =  0 )

Simplify

-3y =  5x + 12

x intercept                             y intercept

0 = 5x + 12                            -3y = 12

-12 = 5x                                    y =  -4

x = -5/12

Distance between  x int and  y  int = 

sqrt [  (-5/12)^2 + (-4)^2 ] = 

sqrt [ 25/144  + 16 ]  =

sqrt [ 2329 / 144  ]   =

sqrt [ 2329  ] /  12   ≈   4.02

6x + (3/5)y

xy = 2/5

y = 2 /(5x)

6x + (3/5)(2) / (5x)

6x  + (6/25)x^(-1)           take the derivative and set to 0

6 - (6/25)x^(-2)  =  0    multiply

6x^2  = (6/25)

x^2 =  1/25

x =   1/5

Take the scond derivative to see that this is a  min

 2(6/25)(.2)^(-3)  =  60   which is > 0 ....so this is a  min

Min  at (.2, 2)    =  6(.2) + (3/5)(2)  =  1.2 + 1.2  =  2.4

(-4,0)  and (0,2)  are on the line

Slope   = [ 2 - 0  ] /  [ 0 -  - 4]  =  2/ 4   =1/2

Equation  

y = (1/2)x + 2          multiply through by 2

2y = x + 4

x - 2y =  -4

OK...i'll try to think about it a litle more

Simplify  as

x^2 - 21x + y^2 + 13y  =  25     complate the square on x, y

x^2 -21x + 441/4   + y ^2 + 13y + 169/4 =   25 + 441/4 + 169/4

(x - 21/2)^2  + (  y +13/2)^2 =  355/2

This is a circle centered at  ( 21/2 , - 13/2)  with a radius of sqrt [ 355 / 2 ] 

Largest value of x =  21/2  + sqrt [ 355 / 2 ]  =   21/2 + sqrt [ 710 ]  / 2

Here's what I get

Planet Y  travels  360 / 450 =  4/5  as fast as Planet X

So.....Planet X  has traveled 360°  in 360 days

And Planet Y has traveled (4/5) * 360°  = 288°  in 360 days

So  there is a 72°  difference between them

So it takes   360 / 72   = 5   360 day periods before they are on the same line again

Planet  Z  travels  360 / 540  = 2/3  as fast as Planet X

Planet  Z travels  (2/3) 360  = 240°  when Planet X travels 360°

So....there is a 120° difference  between them

So...it  takes   360 / 120 =  3   360 day periods   before they are in the same  line again

5 * 3 =    15  360 day periods =  5400 days

Take the derivative with respect to  and  set to 0

2x -16 = 0

x = 8

Take the derivative with respect to y and  set to 0

2y + 16 = 0 

y = -8                            

The coldest temp is

  (8)^2 + (-8)^2-16(8) + 16(-8) + 26  =  -102