CPhill

Note

19^2  =  361

So....a +  b  =   21

We have that

a1 + (3)d =  129    →   a1 =  129 - 3d  (1)   

Where a1 is the first term and d is the common difference

And since the sum of the four angles of a quadrilateral  = 360°

a1  + (a1 + d) + (a1 + 2d) + 129  = 360    (2)

Subbing  (1)  into (2).....we  have

129 - 3d + ( 129 - 3d  + d) + ( 129 - 3d + 2d )  + 129  = 360     simplify

516 - 6d  = 360       subtract 516 from both sides

-6d = -156          divide both sides by  -6

d = 26

And   ...     a1  =  129 - 3(26)  = 51°

So.....the second largest angle is   51 + 2*26  =  51 + 52  = 103°

Here's the second one

We have  that

ceiling (x)  +  x  =  23 / 7     .....where the "ceiling" function represents the smallest integer ≥ x

Note that if   x  =  9/7  we have that 

ceiling (9/7)  +  9/7  =

ceiling ( 1 + 2/7)  +  9/7  =

2    +  9/7   =

14/7 + 9/7  = 

23/7

"r^2"   means that the radius is "squared"

So, for instance, if the radius is 3, the square is 3^2  =  9

By the theory of intersecting chords.....

XV * ZV  =  WV * YV     ....so....

4 * 9  =  3 * YV       divide both sides by 3

[ 4 * 9 ]  / 3  =   YV

36 / 3  =  YV   =  12

And since the chords are perpendicular, by the Pythagorean Theorem.......

ZV^2 + YV^2  = YZ^2

9^2  +  12^2 = YZ^2

225   =  YZ^2        take the square root of both sides

15  =  YZ 

2/3 (s) + 1/3  =  3/4(s) - 1/4      

multiply through by the common denominator of 3,4  = 12

8s  + 4  =  9s  - 3         add 3 to both sides, subtract 8s from both sides

7  =  s

__   2      7        nissing digit =  4

 8   __     6        missing digit  = 3

1     4     __       missing digit  = 6

__________

_ 4  0      9

Proof

4 2 7

8 3 6

1 4 6

____

1409

Inches per hour  =  700 * 2 * pi (17)  * 60  ≈  4486194 in / hr      (1)

Inches in one mile   =  5280 * 12  = 63360         (2)

Mph  =  (1) / (2)  =   4486194 / 63360   ≈  70.8 Mph

___   2      5         missing digit  =  3

  8    __     8         missing digit  =  1

  1     6     _          missing digit  = 4

___________

_3    0     7          missing digit  = 1

Proof

 325

 818

 164

____

1307

 19^2   =  361

Each floor  takes  [ 51] / [ 3 - 1]  =   51/2  =  25.5  seconds

So....... from the first to the sixth - 5 floors -  takes     5 * 25.5  = 127.5 seconds 

Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 40. What would her answer have been if she had worked the problem correctly?

[ N - 9] / 3  =  40   multiply both sides by 3

N - 9  =  120       add 9 to both sides

N  = 129

So....the correct answer should be

[ 129 - 3 ] / 9  =   126 / 9  =   14