CPhill

Circle S is a translation of circle T, 5 units down.

Circle S is a dilation of circle T with a scale factor of 9.

Circle 2 is a dilation of circle 1 with a scale factor of 5.

sin (45)   =  4√2 / x       

1/√2 = 4√2 / x      rearrange  as

x =  4√2 /  [ 1/√2]

x  = 4√2  * [ √2 / 1 ]

x  = 4√2 * √2   =   4 * 2   =   8

Pierre and Thomas can be "anchored" together....and they can be seated in 2 ways

Rosa  can sit in any of the 4 positions not next to Pierre and Thomas

And the other 5 people can be seated in 5!  =  120 ways

So.....the possible seating arrangements are

2 * 4 * 120  =   

960

Volume  =  L * W * H  = 70

But L  = 2W

So we have that

2W * W  * H   =  70

2W^2 * H  =  70

W^2 * H  =  35.......solve for the height

H  =  35/W^2

The surface area  is  the top  and bottom  = [ 2 * L * W]  = [ 2 *2W*W]  =  4W^2

And the total cost per m^2 for this part of the box is $9 * Area  = 9*4W^2  =  36W^2

The surface area of two of the sides  =  2*2W * H  = 2*2W* (35/ W^2) =

140/W

And the surface area for the other two sides = 2*W*H = 2*W (35/W^2)  = 70/W

So....the total surface area of the sides  =  (140/W + 70/W)  =  210/W

And the total  cost per m^2  of this part of the box is $6 * Area = 6*210/W  = 1260/W = 1260W^(-1)

So....the total cost, C,  is  given by

C  =  36W^2  + 1260W^(-1)      take the derivative and set to 0

C'  =  72W  - 1260W^(-2)  = 0

C'  = 72W - 1260/W^2  = 0

72W   =  1260/W^2

72W^3  =  1260

W^3  =  17.5

W  = ³√17.5  m

L  = 2W  =   2 ³√17.5   m

H  =  35 /W^2  =   35 / ³√(17.5)^2   =  35 / ³√306.25   m

The minimum cost  is

36 W^2   +    1260 / W  =

36 ³√[17.5^2]   +  1260 / ³√17.5   =

36 ³√306.25  + 1260 / ³√17.5   ≈   $727.97

This is the intersecting chord theorem......we have that

(n + 1) * 18  =  9 * (n + 7)    simplify

18n + 18  = 9n + 63    subtract 18, 9n from both sides

9n  =  45     divide both sides by 9

n  = 5

f(x)   =  ceiling function  [ 1 / (x + 2) ]    x >-2

f(x)  =  floor function [ 1 / (x + 2) ]  x < -2

Notice that the first function can never  evaluate to 0  when x > -2

And the second function can never evaluate to 0  when x < - 2

So  0  is not in the range of f(x)

[ x^2 + 10x  + 21]

______________

[ x^2 + 4x  -  21 ]  

The domain excludes any x values that makes the denominator  = 0

So

x^2 + 4x  -  21  =  0

(x + 7) ( x - 3)  = 0

Setting each factor to 0 and solving for x  produces the two x values not in the domain  ⇒

x  =  -  7          and  x   = 3

So.....the domain  is

(-inf, -7) U (-7, 3)  U (3, inf)

Note that, by AAS, triangle EC₁D   is congruent to triangle EAB

C₁ D  = AB  =  DC  = 5

BC₁  = BC  = 10

And BD  = √[C₁D^2 + BC₁^2]  = √ [5^2 + 10^2 ]  = √125

Note that  BE^2   = AE^2 + AB^2   =  AE^2 + 25

And  BE  = √[AE^2 + 25 ]

By the Law of Cosines, we have

AB^2  = AE^2 + BE^2  - 2 (AE * BE) cosAEB

5^2  = AE^2 + (AE^2 + 25) - 2(AE√[AE^2 + 25] )cos (AEB)

25  = AE^2 + AE^2 + 25 - 2 (AE √[AE^2 + 25] )cos(AEB)

[-2AE^2] / [ -2AE√[AE^2 + 25]   = cos(AEB)

AE/ √[AE^2 + 25]  = cos(AEB)    (1)

Note that  AEB  and DEB  are supplemental....so  cos(DEB)  = -cos(AEB)

So....using the Law of Cosines again, we have

BD^2  =  BE^2 + BE^2  - 2(BE^2)(-cos(AEB))

125 = 2BE^2 + 2BE^2(cos(AEB) )

125 = 2[AE^2 + 25]  - 2[AE^2 + 25] os(AEB)

125 = 2AE^2 + 50 - 2[AE^2 + 25] cos(AEB)

[75 - 2AE^2[ / [ 2(AE^2 + 25) ]  = cos(AEB)   (2)

Equate  (1)  and (2)

[75 - 2AE^2 ] / [ 2(AE^2 + 25)] = AE/ √[AE^2 + 25] 

[75 - 2AE^2] /(AE^2 + 25)  = 2AE/ √[AE^2 + 25] 

[75- 2AE^2] / (AE^2 + 25)  = 2AE√[AE^2 + 25] / (AE^2 + 25)

75 - 2AE^2  = 2AE√[AE^2 + 25]     square both sides

5625 - 300AE^2 + 4AE^4   = 4AE^2 [ AE^2 + 25]     simplify

5625 - 300AE^2 + 4AE^4 = 4AE^4 + 100AE^2 

5625 =  400AE^2

5626/400 = AE^2

225/16  = AE^2

15/4  = AE

So

DE  =  AD  - AE

DE  =  10  - 15/4

DE  = 40/4  - 15/4

DE  = 25/4   =  6.25

Note  that 

1 + 3 + 5 + 7 +.......+ 199     has  100 terms

And the sum of these =  n^2  = 100^2  = 100 * 100

And  

2 + 4 + 6 + 8 + ......+ 200   also has  100 terms

And the sum of these  is  n(n + 1)   =  100 (101)

So

[  1 + 3 + 5 + 7 + ....+ 199 ]            100 * 100                  100

______________________  =  ___________   =          ___ 

[ 2 + 4 + 6 + 8 +......+ 200]             100 * 101                  101