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(4.5 x 10¹⁵ ) / ( 6 x10⁵) =  ......... years

cos(a+b) = cos a cos b - sin a sin b

cosxcos60 - sinxsin60 = 2 sin x

1/2 cosx - sqrt3/2 sinx = 2 sinx

1/2 cos x = 2+ sqrt3/2) sin x

cos x = ( 4 + sqrt3)sinx

tan x = 1/ (4 + sqrt3 )

x = arc tan[1/ (4 + sqrt3)] = 9.896 degrees

Thank you very much for your help. Yes I guess I am lucky but only on this site. Nobody answered when I wrote on other sites.

https://web2.0calc.com/questions/find-all-complex-numbers-z-such-that-z-4-4-please-help-and-tysm

This has been asked about a half of a dozen times in the last 6 months......search to see if you can find the explanation of the answer...

    here are TWo of th answers

-1+i

-1-i

The base of this  triangle will be  the altitude  of an equilateral triangle with a side of   1 =   sqrt (3)/2

Sides of triangle   =  two sides  =  sqrt (3)/2  + a side of 1

Semi-perimeter of  triangle =   [ 1 + 2 sqrt (3)/2 ] /2  =   1/2 + sqrt (3)/2   = S

Area =   sqrt  [ S  (  S -1) ( S - sqrt (3)/2)  ( S - sqrt (3)/2)  ] 

Area =   

 sqrt  [ (1/2 + sqrt (3)/2)  sqrt (3)/2 - 1/2) (1/2)^2 ] =

sqrt [ (-1/4 + 3/4 ) ( 1/4) ]  = 

(1/2)sqrt (1/2)  = 

(1/2) sqrt (2) /2  =

sqrt (2) / 4   units^2

The line  has a slope of  -2

The  y intercept is  78

The x intercept  is    0  =  -2x  + 78   →    2x  =78    →  x   = 39

So.....x int =  (39,0)   and  y int =  (0, 78)

The line will pass through  a lattice point when x   = any integer  from  0 to 39 inclusive

The number of lattice  points this line will pass  through  is given by

(39  - 0 )  +  1   =

40  lattice  points

Rearrange as

y^2 + 5y   -x^2    = 25     complete the square on y

y^2 + 5y  + 25/4   - x^2  = 25   +  25/4

(y +5/2)^2   - x^2  =   125/4        divide through by   125/4

( y + 5/2)^2 / (125/4)   -  x^2 / (125/4)   =  1

This is a hrperbola  that opens  up /down   

x^2 -2y^2  - 6x  -8y   =   -481

x^2 - 6x  - 2y^2 - 8y  =  -481     complete the square on  x,y

x^2 - 6x + 9   -  2(y^2 -4y + 4)  =  -481   + 9   -  8

(x - 3)^2   -2 ( y -2)^2  =  -480          divide through by 2

(x - 3)^2 /2  - (y - 2)^2  =   -  240      rearrange as

(y -2)^2   - ( x - 3)^2/2   = 240          divide  through  by   240

(y - 2)^2/240  - (x  - 3)^2/480  =   1

This is a hyperbola that  opens  up/down

Semi-perimeter of  triangle   =  (17 + 8 + 10) / 2  =  35/2  = 17.5  =S

Let the sides  be A, B ,C

Area  =

sqrt   [ S ( S - A) (S - B) ( S - C)  ]    so we have

sqrt  [ 17.5  ( 17.5 -17) (17.5 - 10) ( 17.5 -8 ) ]  =

sqrt  [ 17.5 * .5  *  7.5  *  9.5 ]  =

sqrt   [ 623.4375  ] ≈ 

24.97 units^2

1. When y = 10 and z = 3, x = 4.

2. The volume of a cone having a radius 15 inches and a height of 7 inches is 158*pi.

The numerical value of x is 2/3.

There are 4027 triples of integers.

The solution is z = 2 - 7i.

**** edit ***

= 2 ( x^2-3x)       + 8

= 2 (x-1.5)²   -4.5 +  8

= 2 (x-1.5)2  + 3.5             take it from here................

= 2 ( x^2-6x)       + 8

= 2 (x-3)²   - 18 + 8

= 2 (x-3)²  -10             take it from here................

[6, +inf)

its 4

4*3=12

12/3=4

(a) Remember that to prove that a number is divisible by 6, you just have to prove that it is divisible by 2 and 3.

(b) The solultions are n = 2 and 7 (mod 12).

The length of CE is sqrt(87).

By the Sine Law, XY = 2*sqrt(2) + 2*sqrt(3).