Question1.
(1,1) → (6,7) → (2,7) → (2,3) → (7,9) → (3,9) → (3,5) → (3,1) → (8,7) → (4,7) → (4,3) → (9,9) → (5,9) → (1,9) → (1,5) → (1,1)
15 different points.
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From Wikipedia:
"Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships involving lengths and angles of triangles."
I don't think there's a strong reason for expressing the answer this way. You could have asked the same question if it had been expressed your way! Sometimes it is easier to do further derivatives if the results are given in one form rather than another, but in this case either form is easy to take further it seems to me. Certainly, there is no general rule I'm aware of that specifies what form such a derivative should take.
Like so:
A brute force and ignorance calculation shows you to be correct Chris.
There are 91 numbers in the range that are divisible by both 9 and 11
However, the sum of their digits is 18 for all but one of them (the odd one out is 9999, the digits of which sum to 36).
The digital root of each of the 91 numbers is 9 of course. (For the digital root you keep summing the digits until you are left with a single digit.)
6 divided by 3/8 is the same as 6 multiplied by 8/3: 6*8/3 → 2*8 → 16
These two expressions are the same:
ake-kx(kx-2)
ae-kx(k2x-2k)
If you factor a k out of the parentheses in the second expression you get the first expression.
Set y = 61 and find x
61 = 14.7x + 16.9
x = (61 - 16.9)/14.7
x = 3
So spending will be more than $61 billion in all years after (2002 + 3), I.e. from 2006 onwards.
Use the calculator on the home page here.
In 3d with points (x1, y1, z1) and (x2, y2, z2) the distance formula is:
distance = sqrt( (x1-x2)^2 + (y1-y2)^2 + (z1-z2)^2 )
You need three coordinates for each point!
+33667 
