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The  end point will be  ( -1/2, -√3/2 )

The coordinates of the point that intersect the unit circle are (x,y)

According to the Pythagorean theorem:

x² + y² = 1²

x² + y² = 1

According to the problem statement:

y/x = -2

So,

y = -2x

Substitute:

x² + (-2x)² = 1

x² + 4x² = 1

5x² = 1

x² = 1/5

x = ±√(1/5)

x = ±√(5) / 5

y = -2[ ±√(5) / 5 ]

y = ∓2√(5) / 5

The points are: \((\frac{\sqrt5}{5},-\frac{2\sqrt5}{5})\)    and    \((-\frac{\sqrt5}{5},\frac{2\sqrt5}{5})\)

The line has the equation  y = -2x      (1)

The circle has the equation  x^2 + y^2  = 1     (2)

Sub (1)  into (2)

x^2 + (-2x)^2   = 1

x^2 + 4x^2  = 1

5x^2  =  1           divide both sides by 5

x^2  =   1/5           take both roots

x  = ± 1/√5

And using (1)   when   x  = 1/√5,   y  = -2/√5

And when  x = -1/√5,  y  =  -2 (-1/ √5)  =   2/√5

So.....the intersection points are   ( 1/√5, -2 /√5 )   and   ( -1/√5,  2/√5)

m/2-5m/6=7    multiply through by the common denominator of 2, 6  →  6

3m - 5m   = 42

-2m   =  42        divide both sides by -2

m = -21

because it is

Your result doesn't take into account the fact that half the matrix is set to zero:

     000000000000
     000000000001
     000000000034
     000000000400
     000000006868
     000000048049
     000000536107
     000007379780
     000056289484
     000869347589
     005626115408
     042828679713
     679952567873

So the correct result is 10^12 rows times 12 columns, divided by ten (limit to a single digit in the set {0-9}) and divided by two to account for the empty half of the matrix = 6 * 10^11 threes.

One problem. The array of numbers, 10^12 rows high and 12 digits wide, is half zeros:

     000000000000

     000000000001

     000000000034

     000000000400

     000000006868

     000000048049

     000000536107

     000007379780

     000056289484

     000869347589

     005626115408

     042828679713

     679952567873

Therefore the correct result is 1 * 10^12 (height) * 12 (width) / 10 (limit result to threes) and finally / 2 (to account for the empty half of the matrix) = 6 * 10^11

We can use the tangent, here

tan (240)  = tan(60)   =  √3/1  = √3

And that will be the slope of the radius

We can use similar triangles here

30 / 52   =  H / 20        where H is the height that the shadow stries the building

Cross-multiply

20 * 30   = 52 * H

600 = 52 * H    divide both sides by 52

600 / 52  = H  ≈  11.5  ft 

Here's a pic :

Here's a nice trick:

There's a way to find every single rational root for some of the polynomials:

step one:

we have the polynomial a₀+a₁*x+a₂*x²+a₃*x³+........+a_n*xⁿ=0. We need to multiply the polynomial by a number in a way that all coefficients will be whole numbers. Sometimes we cant do it. sometimes we can.

step two:

find all of the whole divisors of a_n and a₀ (including the negative divisors). lets call the set of the divisors of a₀ A and the set of the divisors of a_n B. there are k divisors in A and m divisors in B.

Every rational root of the polynomial=A_i/B_j, 0 i is a member of A and B _j is a member of B, meaning every rational root can be expressed using one divisor from A divided by one divisor from B).

step three:

Check for every two divisors (one from A and one from B) if A_i/B_j is a root. after a finite amount of time (unless your calculator is broken) you'll have every rational root of the polynomial.

ab^3  =  54         a = 2     b = 3

2 * 3^3  =

2 * 27  =

54

Solve for x:
x^3 - 7 x^2 - 12 x + 54 = 0

The left hand side factors into a product with two terms:
(x + 3) (x^2 - 10 x + 18) = 0

Split into two equations:
x + 3 = 0 or x^2 - 10 x + 18 = 0

Subtract 3 from both sides:
x = -3 or x^2 - 10 x + 18 = 0

Subtract 18 from both sides:
x = -3 or x^2 - 10 x = -18

Add 25 to both sides:
x = -3 or x^2 - 10 x + 25 = 7

Write the left hand side as a square:
x = -3 or (x - 5)^2 = 7

Take the square root of both sides:
x = -3 or x - 5 = sqrt(7) or x - 5 = -sqrt(7)

Add 5 to both sides:
x = -3 or x = 5 + sqrt(7) or x - 5 = -sqrt(7)

Add 5 to both sides:
Answer: | x = -3      or x = 5 + sqrt(7)      or x = 5 - sqrt(7)

You can go to this link and download as many digits of Pi as you want, up to a trillion I believe.

And while you are there, you can download their program to calculate Pi on your own computer. It is the fastest such program on the internet. Among other things, it will test the speed of your computer as well! As a reference, I downloaded it and computed 25 million digits of Pi in 2.505 seconds on my laptop. Here is the link: http://www.numberworld.org/y-cruncher/

The way to approach this problem is as follows:
Write each number as 12 places, as Ehrlich did, as follows:
000,000,000,001, 000,000,000,002, 000,000,000,003...and so on.
Since we have 10^12, or a trillion numbers, and each one of them consists of 12 digits, we, therefore, have a TOTAL OF: 12 x 10^12 =12 trillion INDIVIDUAL DIGITS. 
And since each of the 10 digits(0 - 9) has an equal representation in the above 12 trillion individual digits, we therefore will have:
12 trillion(or 1.2 x 10^13) digits / 10 =1.2 trillion =1.2 x 10^12 threes in 10^12 numbers. 

How did you get that? I did it and only got 3.1415926535897932!

Good Job! Nice explanation, bu

.

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x/360 * 2pi*r = 7pi*r/10

Solve for x:
(π r x)/180 = (7 π r)/10

Divide both sides by (π r)/180:
Answer: | x = 126

2-x=4+x

-x -x = 4 - 2

-2x = 2

x =-2/2

x = -1

8x + 13 = 7.5x + 16

8x - 7.5x = 16 - 13

.5x = 3

x = 3/0.5

x = 6

6638 elves

LOL..Now you made me check a third time!!!

8 + 3   IS  11 ????......heck.....I've been doing that wrong my whole life......!!!!

Lol that's for sure!

A lot of times I have to double-check things like 8 + 3, just to make sure it's still 11..haha you never know when it might change up on you one of these days!! x)

Don't worry....we second-guess ourselves a lot  on here, too....!!!