Planting trees

[Fusion_power]

So there is a man who likes to plant trees. He's really picky about how the trees are planted and wants them to have the very best possible conditions to grow in but at the same time wants to put the most possible trees on the least possible area.

After due consideration he determines that if he plants the trees in an equilateral triangle with a tree at each of the vertices of the triangle and repeats this pattern so that a single tree in the middle is surrounded by 6 equally spaced trees, he will have achieved his objective. Next he figures out that the trees will grow to their optimum if they are spaced exactly 30 feet apart.

Presume that the roots of one tree grow precisely half way to the next tree and vice versa

Now the somewhat tricky question. What area in square feet will each tree have to grow in?

No awards for fancy formulas, this one is really simple and easy to calculate.

Fusion

[Fusion_power]

As I thought, there's nobody here who can figure out the area of a hexagon!

I guess the upgrade to a new server was more damaging than I originally suspected. In this case, most of the mathematically competent people seem to have been left on the old server. I wonder what they do there? It must be something like Robin Broadhead in the HeeChee saga. They're living in digital time and don't have time for mundane affairs.

I think I'll go out and plant a few more trees. First measure out one long straight base line. Then measure a second line at a 60 degree angle. Next stretch as many lines as required parallel to the baseline. The intersection of the 60 degree line with the parallel lines defines a single planting point. Plant trees every thirty feet from there along the parallel line.

Resulting pattern is a simple hexagon with a tree in the middle.

Fusion

[Wacky]

Fusion_power --

Did you read the PM that I sent you a few days ago?

As you note, if you plant trees on an equilateral triangular grid and assign each point on the ground to the nearest tree, the area assigned to each tree forms a hexagon around the tree where the lines joining adjacent trees are the perpendicular bisectors of the sides of the hexagon.

Now, there is a tree at each vertex of the triangle, and everything is symmetric. So 1/3 of each triangle is assigned to each of its trees. But since there are six triangles associated with any given tree, the total area associated with that tree is twice the area of one triangle.

The triangle is equilateral. Its height is 0.5* base * sqrt(3) and its area is 0.25 * base^2 * sqrt(3).

Therefore the area associated with each tree is 0.5 * base^2 * sqrt(3).

In your problem, the base is 30 ft. Therefore the area is 450 * sqrt(3) sq.ft. or approximately 779.4 sq ft.

[Fusion_power]

Indeed Wacky I did get your msg as well as from Maybeso and NickGlover.

Each of you arrived at the right answer but at the expense of complicated math.

The first time I solved this I went through similar machinations.

Here's the simple form.

A = D^2*.866

where D = 30 (distance from one tree to next)
where .866 is the sine of 60

900*.866 = 779.4

You could also look at D as the height of the hexagon.

Fusion - "out of the box" thinker of sorts

[Wacky]

Your initial response failed to acknowledge any of our responses. I think that you should have had the courtesy to acknowledge that we had presented the correct answer and then offered your "simpler" solution.

Ignoring the above, I take exception to your answer because it effectively just states an answer without any supporting background. I cannot take for granted that anyone knows the formula for the area of a hexagon. So my "complicated math" was really just the derivation of the formula which you assumed everyone knew.

How does your answer make any more sense than an answer such as "The number is a Paulie Number and therefore the answer is 42"?

[Fusion_power]

Lighten up a bit wacky,

Note the line in the original post, "No awards for fancy formulas". I was really curious how many people would get past the sqrt 3 and see the relationship to sine 60. My post was intended to prompt someone to respond, not to infer anyone was incapable of doing simple math.

Why else would I refer to Robin Broadhead etc. Why would you refer to the number 42 if you had not read Hitchhiker's Guide to the Galaxy? BTW, I finally figured out the "ultimate question" but it got lost in an episode with Marvin the paranoid android.

Fusion

[wblipp]

Originally posted by Fusion_power
Why would you refer to the number 42 if you had not read Hitchhiker's Guide to the Galaxy?

Because you are a Lewis Carroll fanatic.

From the Hunting of the Snark:
Rule 42 of the Code , "No one shall speak to the Man at the Helm."

He had forty-two boxes, all carefully packed, ....

From Alice's adventures in Wonderland:
Rule 42: All persons more than a mile high to leave the court.

Also Phantasmagoria and a dozen or so more labored metions.

[Wacky]

Originally posted by Fusion_power
Note the line in the original post, "No awards for fancy formulas". I was really curious how many people would get past the sqrt 3 and see the relationship to sine 60.

Sorry, but I was always taught that an answer involving "trig" was not simpler. Similarly, a solution that involves calculus is even more complex.

Why would you refer to the number 42 if you had not read Hitchhiker's Guide to the Galaxy?

Bad assumption. I have never seen a copy of the Hitchhiker's Guide, much less read it.

As wblipp notes, "42" may have been popularized by the Guide, but not have originated there. There are many examples in literature where one author's work intentionally makes reference to something from some prior work. Understanding these allusions is a part of the reader's enjoyment.

[xilman]

Originally posted by Wacky
As wblipp notes, "42" may have been popularized by the Guide, but not have originated there. There are many examples in literature where one author's work intentionally makes reference to something from some prior work. Understanding these allusions is a part of the reader's enjoyment.

It is only to be expected that the answer to the Ultimate Question of Life, the Universe and Everything should be widespread in the memory banks of a computer in the final stages of computing the question shortly before the entire computer is destroyed.

Personally, I believe the answer is entirely fortituitous and was influenced by irresponsible usage of Infinite Improbability drives.


Paul

[Jwb52z]

It just makes you wonder where they got all the tea, ya know.