CPhill

https://web2.0calc.com/questions/geometry_93163

https://web2.0calc.com/questions/right-triangles_33

Euler's Totient Method

Prime factors of 2835 =  3 , 5, 7

No. of Positive integers relatively prime to 2835  =

(2835) / (3 * 5 * 7) * ( 3 -1) (5-1) (7 -1)  = 

27 (2) (4) (6)  =

1296

13/12 =  1/1  + 1/12  =  1/1  + 1/13 + 1/156

The first fraction is  easy....use the "Egyptian Method" to  generate the last two....this will give us fractions with numerators  =1

Second fraction  denominator = invert the second fraction  and  add 1   =  12/1 + 1 =  13

Second fraction =1/13

Subtract  1/12 -1/13  to get the last fraction

112 -1/13  = [ 13 -12] / 156   =1/156

3 hrs 20 min  = 3 + 1/3 hrs = 10/3 hrs 

3 hrs 50 min    =  3  + 5/6 hrs =  23/6 hrs

Total time = 43/6 hrs

Let the distance between the towns = D

Let the speed of  the plane  with no wind = S

Let  the speed of the plane against the wind = S - 70

Distance  / Rate = Time

With no wind

D / S = 10/3

D = (10 / 3) S

With wind against the plane

D / (S -70) = 23/6

(10/3)S =  (23/6) (S -70)

(10/3)S = (23/6)S  - 1610 / 6

(23/6 - 10/3)S = 805 / 3

(23/6 - 20/6)S = 805/3

(3/6)S  = 805/3

(1/2)S  = 805/3

S = 1610 / 3

Distance =  Rate * Time =  (1610/3)(10/3)  ≈ 1788.9 miles

https://web2.0calc.com/questions/angles_119

https://web2.0calc.com/questions/hexagon_12

Logic :

5 members must take  twice as long as 10 members  = 20 min

Number of diagonals in a regular polygon with n sides = (n)(n -3) /2

So

  [(n) ( n -3) / 2] =  2n

(n) ( n-3)  = 4n

n^2 - 3n = 4n

n^2 - 7n  =0

n ( n -7)  = 0

n = 0   reject

n = 7

The slope of the given line  = -1

The slope of a line tangent to the circle is  given by

-x/ y

So  setting this  = to  the slope of the line

-x / y   = -1

x = y

So

x^2 + x^2  = k

2x^2  = k

x^2  = k/2

x = sqrt(k/2)

So

x + y =3k

sqrt (k/2) + sqrt (k/2) =3k

2sqrt (k/2)  = 3k           square both sides

4k/2 = 9k^2

2k = 9k^2

k = (9/2)k^2

(9/2)k^2 - k = 0

k ((9/2)k -1)  =  0

k = 0   reject

k = 2/9

3k = 6/9 = 2/3