CPhill

0.7......    

First....find the fractional representation of this.....

Take  the  repeating part = 7   

This is the  numerator of our fraction

For the denominator,  write the  number of  9's   equal to  the number of digits in the  repeating part =  one  9

So....  0.7.......  =   7/9

The reciprocal  = 9/7

AC = AB  = 1

CY  =   AY - AC = 6 - 1 =  5

Area of square ASHY  = (AC + CY)^2  =  (1 + 5)^2  =   6^2   = 36

Area of  2 smaller squares = 2 (1)^2 =  2

Triangles  CDY,  YEG, EFS  and SDB  are congruent,,,,each has a base of  5 and a height of 1

So....their combined area =  4 (1/2) (5) (1)   =  10

Area of DYES  = 

36 - 2  - 10   =   24

The diagonals of a parallelogram will bisect  each other....so we have

PT = TR       and   QT = TS

So...we have  this system

y  = 4x + 1       (1)

2y = 5x + 8      (2)

Sub (1)  into  (2)   and we have that

2 ( 4x + 1)  = 5x + 8

8x  + 2   = 5x +  8

8x - 5x  =  8  - 2

3x   =    6    →    x  =  2

And

y = 4(2)  +  1  =   9

We need to observe  the triangle inequality rule that says   that  the  sum of the lengths of any two sides of a triangle  is greater than the  length of the  remaining side

So we have

5 + 10  >  x

15 >  x

So....the  largest integral value of  x =  14

And

5 + x  >  10

x >  10-5

x > 5

So....the smallest integral value of x =  6

Sum of   intergers from  6 - 14  inclusive  =

[first integer + last ineger ]  * [ number of terms ]   / 2   =

[6 + 14]  [ 14 - 6 + 1 ]   / 2   =

[20 ] [ 9 ] / 2  =

(20/2)  * 9  =

10 ( 9)  =

90

The first card is a given

On the second draw, we have  a (39/52)  = (3/4)  probability of drawing a different suite from the first draw

On the third draw we have  a (26/52) = (1/2) probability of drawing a suite different from the first two draws

On the last draw we  have a (13/52) = (1/4) probability of drawing a different suite from the ones we  drew on the first three draws

So  the probability =    ( 3/4) ( 1/2) ( 1/4)  =   3 /32

The radius of the  circle can be found as

18pi  = 2pi * r

18pi / (2pi)  =  r   = 9

We can divide the  equilateral triangle into three congruent triangles.....the area of  of of these triangles is given by

(1/2) r^2  *sqrt (3) / 2  =    (1/2) 9^2  *  sqrt (3)  / 2  =   81 sqrt  (3) / 4  cm^2

So....the area of the equilateral triangle  =    3 (81)sqrt (3) / 4  cm^2  =  (243)sqrt (3) / 4  cm^2

Probably an easier way to do this....but.....

Let the bottom left vertex =  (0,0)  = A

Let the bottom right vertex =  (8,0)   = B

Let the  top left vertex  =  (0,10) = E

Let the top right vertex  ( 8,12) = C

Let the apex vertex =  D

CE =   sqrt ( 8^2 + 2^2)  = sqrt  ( 68)  = 2sqrt (17)

Semi-perimeter  of triangle   CED  =  ( 4 + 6 + 2sqrt (17) ) / 2 =    5 + sqrt (17)

Area of triangle   CED  =

sqrt  [ (5 + sqrt (17) )  ( 5 + sqrt (17) - 2sqrt (17)) (5 + sqrt (17) - 6)  ( 5 + sqrt (17) - 4 )  ]  =

sqrt [ (5 + sqrt (17) ) (  5 -sqrt (17)) ( sqrt (17) - 1) ( sqrt (17) + 1)  ]   = 

sqrt [ (5 + sqrt (17) ) (5-sqrt (17) ) (sqrt (17) -1) (sqrt (17 + 1)  ]   =

sqrt  [ (25 -17)  ( 17 - 1)  ]  =

sqrt [  8 * 16 ]  =  

sqrt ( 64 * 2)  =    8sqrt (2)

And  ABCE  is a trapezoid  with  bases of  10 and 12  and a height of 8

[ ABCE ] =    (1/2) [ 8] [ 10 + 12 ]=    4 * 22   =  88

So...area of the pentagon  = [  88  +  8 sqrt (2) ] in^2  =    8 (11 + sqrt (2))  in^2

If we  have  the points  A=  ( -8,0)  and C =   ( 8,0)  on the circle....then AC will form a chord in the  circle

This means that  the center  of  the  circle  will  have  the coordinates  (0,a)

And  the distance  from the center  to   B  will be the same distance from   the center to   C

So  we have this

(-3)^2  + (4-a)^2  =  (8)^2  + ( a)^2 

9 +  16  - 8a  + a^2  = 64 +  a^2     simplify

-39  = 8a

a  =  -39/8

The center is  (0, -39/8)

The radius ^2  =    ( 8)^2  +  ( -39/8)^2 =   5617/64

The equation is

x^2  +  ( y + 39/8)^2  = 5617/64

  See the graph here :  https://www.desmos.com/calculator/ul4b4frbtz

One hour  = 3600 seconds

So

y =  1.5 (3600) =  5400  meters

I think you have something like this  :

There is an interesting relationship here between the base angles and  the "wing"  angles  at  C and E

These  two angles  =  base angle measure  / 3 =    105 / 3 =   35°

FYI  -   the sum of the interior  angles = 540°

So....the  angle at D  =  540  - 2( 105 + 35)  =  540 - 2 ( 140) = 540 - 280  =  260°