CPhill

Let XW and VY intersect at Z

Inscribed angle VXW  intercepts minor arc VW....so its measure is  85/2  =  42.5

Inscribed angle XVY  intercepts minor arc XY.....so ys measure is  162/2 =  81

So angle XZV = 180 - 42.5 - 81  = 56.5

By the interecting chord theorem

Measure of angle XZV = (1/2) ( minor arc WY  +  minor arc XV)

56.5  = (1/2) (minor arc WY + minor arc XV)

113 =  minor arc WY + minor arc WV       (1)

By the theorem of  the angle formed by two secants we have that

Measure of angle U = (1/2) (minor arc WY - minor arc XY)

27 = (1/2) ( minor arc WY - minor arc XY )

54 =  minor arc WY - minor arc WV   (2)

Add (1) , (2)

167  = 2 minor arc WY

minor arc WY =  167 / 2  =  83.5°

The average will be    2^n / (n + 1)   where n is the row number =  0,1,2,3........

So

2^(n )  / (n + 1) =  2

2^(n)  = 2(n + 1)

From inspection, we can determine that  n  =   3 

            1

      1        1

   1      2        1

1     3       3       1   →    3td  row

LOL!!!   Thanks !!!!

                                                                  Resulting situation                  

2 C row over 1 C comes back      →         1C 2A         1C

1A rows over 1 C comes back      →         2C 1A         1A

2C  row over  1 A comes back      →         2A               2C

1A  rows over 1C comes back      →        1A 1C         1C 1A

1A rows over 1C comes back       →          2C              2A

2C row over                                   →          0                 2C 2A

15 / .3048  *  60 / .3048 ≈  9688   sq ft

Ceiling [  9688 /  400 ] =   25 gallons

 yz

____   = 4     →   yz = 4( y + z)  →  yz - 4y = 4z  → y (z-4) = 4z  → y = 4z / (z - 4)      

y + z

  xz

____  = 2  →  xz = 2(x + z) → xz- 2x = 2z  → x ( z - 2) = 2z → x = 2z / (z-2)   

x + z

xy

_______ = 1   →   xy  = x + y       (1)

x + y 

So, using (1)

4z / (z -4)  *  2z / (z -2)  =    4z / (z-4)   +  2z/(z -2)           mulyiply through by (z -4) (z-2)

4z * 2z  =  4z (z - 2) + 2z (z-4)

8z^2   =  4z^2 - 8z + 2z^2 - 8z

2z^2 + 16z  = 0

z^2 +8z  = 0

z ( z + 8)  = 0

z = 0   (reject)

z =  - 8

Nice answer, NotThat Smart !!!!

First one

Cross-multiply and  we  get

2x √(6x) + 4√x  = x√3 + √2       square both sides

4x^2 (6x) + 16x  √(6x^2) + 16x  =  3x^2 + 2x√6 + 2

24x^3 + 16x^2 √6 + 16x  = 3x^2 + 2x√6 + 2         factor the left side

8x ( 3x^2 + 2x√6 + 2 ) =  (3x^2 + 2x√6 + 2)          divide both sides by (3x^2 + 2x√6 + 2)    

8x  =  1

x =  1/8 

Let B = (0,0)   C  =(32,0)

To find the height of the triangle we can construct two lines  using the tangents 

y = (3/5)x

And

y = (-1/4) (x -32) =  (-1/4)x + 8

To find the x coordinate   of their intersection

(3/5()x =(-1/4)x + 8

(3/5 + 1/4)x =  8

[(12 + 5] / 20]  x  =  8

[17/20] x = 8

x = (8*20] / 17  =  160/17

The height of the triangle (AD)  = (3/5)(160/17)  = 480 / 85 = 96/17

Area ABC  = (1/2) BC * AD =  (1/2) 32 * (96/17)  = 1536 / 17

sinA =  1/3

So

BD / AB  =  1/3

BD / 24  = 1/3

BD = 24 (1/3)  = 8

sin C = 1/2

Angle C  = arcsin (1/2)  = 30°

Then triangle BDC  is a 30-60-90  right triangle

BD  = 8

So  BC   =  2BD  = 2 * 8  =   16