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Question
An acre of land is to be planted with trees.  The trees are to be placed 4 feet apart.  Determine number of trees.

Note: The acre is of various shapes - square, rectangle, circular.

Thanks.

Answer
Note that I am not at all sure that this is correct.

I assume the land is to be planted as densely as possible, but I don't know how close a tree can be to the perimeter (4 ft from the edge? 2 ft? 0 ft?).

1 acre = 43,560 square feet

Since trees are 4 ft apart, each tree must be surrounded by at least 2 ft of space in all directions. Suppose each tree is at the center of a circle of radius 2 ft. The area of the circle 4π ft². The most compact arrangement is when a circle is surrounded by six circles (assuming the perimeter of the land is not an issue).

You might suppose that the number of trees is 43,560/(4π). However, this is incorrect because no matter how compactly you arrange the circles, there are gaps.

So, instead of using circular planting spaces, I used hexagons, because they can be packed together without gaps. To maintain the four-foot spacing, the tree is at the center and the apothem of the hexagon must be 2 ft.

The area of each hexagon is approximately 13.9 ft²:
https://www.flickr.com/photos/dwread/16008665493/

43,560/13.9 ≅ 3143.7
The maximum number of trees is 3143.  

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