CPhill

               A

                         E

 B            D                C

If AB = AC  then the triangle is isosceles

If BC is 15   then AD  =   2 * 200 / 15    =     80/3

And because ABC is isosceles, then  DC = BD  = 15 /2

And since DE is parallel to AB  and AC is a transversal cutting both, then triangles   ABC  and EDC are similar because angle ACD = angle ECD   and angle BAC  = angle   DEC

And  since  BC = 2 DC.. then  2 is  the  scale factor between triagles ABC and EDC.....then the area of triangle  ABC  =  area of triangle EDC * scale factor ^2

So

200 =  area of EDC * 2^2

200 = area of EDC * 4

200  / 4  =  area of EDC  = 50

So  [ ABDE ]  =    area of ABC - area of EDC  =  200 -  50     =  150

The cone's radius =    1/2 the side lengt of  the  cube  = 5

And its height  = the side length  =10

So

Volume   = pi * r^2 * h  / 3   =    pi * 5^2 * 10   / 3  =  (250 pi) / 3   units^3

3f ( n - 2) - f(n - 1)

So

f(1) =

3f (-1)  - f(0)  =  -1

f(2)  =

3 f(0)  - f (1) =   3

f(3) =

3f(1) - f (2)  =  3 (-1) - 3   =    -6

f (4) = 

3 f(2)   - f(3) = 3(3)  -  (-6)  =  15

f(5)  =

3f(3) - f (4)  =  3(-6) - 15  =  -  33

The diagonal across the base =   sqrt ( base area) * sqrt 2  =   sqrt (288) * sqrt 2 =    12 sqrt 2* sqrt 2  = 24

1/ 2 of  this =   12

18 is the slant  height

The true height , h  =   sqrt  [18^2  -12^2 ]   = sqrt [ 324 - 144 ]  =  sqrt 180  =  6sqrt 5

The volume is

base area * height /  3  = 

288 * 6 sqrt 5  / 3  =   

288 * 3sqrt 5  cm^3  ≈   1932 cm^3

Sum of 1st n odds  =  n^2  ......we  have   [ 199 + 1 ]  / 2   =  100  odds

Sum of first evens =    [ first even + last even ] * n  / 2 

We have  202 / 2 = 101 evens

So  [ 204] * [101] / 2  =   even sum =  101 * 101

So

sum of odds  /sum of evens   =

100^2   / [ 102 * 101 ]  =

5000 /5151

French=   .11 * 5000  =   $  550

History  = .21 * 5000 =     $1050

Reference   =  .15 * 5000 =  $750

Science = .15 * 5000 =  $750

Bio =  .11 * 5000 = $550

Geo  = .09 * 5000 = $450

Fic =  .18 * 5000 =

Add the numbers in  the right hand  column as  a  check

550 + 1050 + 750 + 750 + 550 + 450 + 900 =    $5000

ln e    =  x

This   means  we  want  to  find the  value of    e^x  = e

It's obvious that  x =   1

We can find the slope thusly :

slope =  tan  [ (arctan (5)  + arctan(2) )  / 2 ]     =   (  9 + sqrt 130)  /  7   

A

10

B            22              C

If BM  is a median, then CM =   sqrt  [ 10^2 + 22^2 ]  / 2    =  sqrt (584) / 2 = 2sqrt (146) /2 =    sqrt (146)  =    AM

And M will be the circumcenter of the circumradius of triangle ABC....so  we have triangle ABM  where

AM = BM    = sqrt (146)  and AB = 10

Using the Law of Cosines

AM ^2  = BM^2 + AB^2  - 2 (BM * AB)  cos ( ABM)

146 = 146 + 100 =  -2 *sqrt (146)(10) cos (ABM)

-100 / [ -20 sqrt  (146 ) ] =   cos ABM

cos ABM  =   5  /sqrt (146)  =  (5/146)sqrt (146)

x^2  +  y^2  = 2x + 4y + 3x + 9y

x^2  +  y^2   = 5x + 13y

x^2 - 5x  + y^2  - 13x    = 0             complete the square on  x and  y

x^2  - 5x + 25/4  + y^2 - 13x + 169 /4    =  25/4 + 169/4

(x - 5/2)^2  +  (y - 13/2)^2  =  190 / 4

This is a  circle  centered  at  (5/2 , 13/2)  with a radius of  sqrt (190)   / 2

x will be maxed when  y =  13/2

It's largest possible value =   [5 + sqrt (190) ]   / 2   ≈   9.39