Some apples
- How are your apples these days?
- Just fine, thank you. Why do you ask?
- No, no reason. I just came to remember that you have quite a few apple trees.
- I have exactly 20,001 x 20,001 – 1, or 400,040,000 in all.
- How do you know the exact number? Do you count them every day?
- I have put the trees in a square grid with my house in the center.
- Please explain.
- To the left of my house I have a row of 10,000 trees, equally spaced out with ten metres between them.
- And to the right?
- The same. And the same above and below the house.
- Do you grow trees in the sky?
- Very funny.
- And I assume you have trees in rows and columns with ten metres between each row and between each column.
- As I said, they are in a square grid.
- Now I remember why I came to think of your apples.
- Tell me.
- How many of your trees are exactly 10,000 metres from the center of your house?
- I know of four, but I guess there may be more.
- How many more?
February 25th, 2009 at 8:59 am
Let (0,0) be the location of the house.
Start with the house at point (10000,0).
Let a=the angle from the x axis as we rotate the radius of 10,000 m around the origin.
A would then be the distance from the x axis or our “y coordinate”.
B would be the distance on the x axis from the origin or our “x coordinate”.
C is constant at 10,000
Cos(a)=B/C
Sin(a)=A/C
Spreadsheets come in nice and handy for these things. We’re looking for nice round numbers for C•Sin(a).
In the first quadrant, excluding the axes, there are trees at:
9600,2800
9360,3520
8000,6000
6000,8000
3520,9360
2800,9600
x4 quadrants = 24
+4 trees on the axes = 28 total.