Naked Science Forum
General Science => General Science => Topic started by: Ians Daddy on 30/06/2009 07:23:41

My daughter and I were building a raised garden in the yard and we discussed the sizes to build it. After a few different sizes discussed, we both noticed a problem. Now, this may be simple math, but I just can't wrap my brain around this one. Please explain it to me so I can explain it to her. Please bare with me, as I'm a "Yank" and use feet and square feet.
Here goes:
I have two boards that are 8 feet long. If I make the garden square with 4 sides of 4 feet, I have a surface area to plant in of 16 square feet. If we go with cutting the boards at 5 feet with a 3 foot remainder, we have a 3 foot by 5 foot garden. Now we have 15 square feet to plant in. 2 foot by 6 foot = 12 square feet. etc. etc.
How does the surface area change if the lengths of the boards do not. They are only cut differently. Please help us out.
Just a thought.

Whether you are working in metric or good ol' feet and inches (*sob*) Mr. "Yank", the answer is the same....................................... So is the question................ And a damnably good one it is at that!
C'mon all you mathematicians, after reading this question I would like the answer too, it's got me flummoxed. Not that that's difficult to do!

Hmm...now that is a good question. I am sorry, I do not have an answer, only the maths behind it.
If we do some simple algebra, 2L + 2W = 16 simplify this and we get W = 8L
The A(rea) = LW
So A = L(8L)
A = 8L  L^{2}
You will get an upside down parabola with the maximum at (4,16) so the maximum area is when the length is four feet, 16 feet squared. If you go either side of this number, ie, go above or below four, the area decreases.

A square just gives better value in terms of area, you could have a rectangle 40 million feet long but 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 feet wide and you'd have approximately no area, but a 20 million feet by 20 million feet garden would be quite a sight.

Oh my !!..Ians Daddy ! [:o]

My daughter and I were building a raised garden in the yard and we discussed the sizes to build it. After a few different sizes discussed, we both noticed a problem. Now, this may be simple math, but I just can't wrap my brain around this one. Please explain it to me so I can explain it to her. Please bare with me, as I'm a "Yank" and use feet and square feet.
Here goes:
I have two boards that are 8 feet long. If I make the garden square with 4 sides of 4 feet, I have a surface area to plant in of 16 square feet. If we go with cutting the boards at 5 feet with a 3 foot remainder, we have a 3 foot by 5 foot garden. Now we have 15 square feet to plant in. 2 foot by 6 foot = 12 square feet. etc. etc.
How does the surface area change if the lengths of the boards do not. They are only cut differently. Please help us out.
Just a thought.
OH My is right! Welcome back to the forum!

if your boards are 8 feet in length and you cut them to 4 you still have to hammer them together are they 2x4's you will loose approximately 8 inches per side per ft, right, as you have to account for width of the frame 1x4's or 8x1's etc.. and that can change again are you laying them on their flat wide surface or on the two inch sides up the thickness once framed in becomes smaller depending on the thickness of your boards and how you framed them in and put them together, at least I believe thats how it works,Once framed in you measure the inside surface area and not outer board to outer board.

Thanks, guys.
Karen, I haven't built it yet, so it was a question that I just pondered in my wee little brain. That does make sense though that the widths of the boards at the adjoining edges would eat up some of the space. However, even on paper with a pencil thin line it does this space time morphing thing.
Still perplexed.
Just a thought.
PS Good to be back

Mad Scientist is almost right [;)] but a 40 million x 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 ft plot wouldn't fit in a 20 million x 20 million ft garden.
What you're doing is confusing perimeter with area. You'd get optimum area by bending the two planks into a circle.
circumference = π * diameter, so the diameter for a circumference of 16 ft (the total length of the two boards) = 16 ft / π = 5.0929 ft
The area of a circle = π * radius^{2} = 20.3718 sqft.
I doubt you'll be able to bend the boards into a perfect circle though, so cut them into 12 or six inch lengths to form a 16 or 32 sided polygon, and to help you put them in place, just mark out a five ft diameter circle to use as a guide.

Mathematics is the only subject where you could get away with saying "You'd get optimum area by bending the two planks into a circle."

Mathematics is the only subject where you could get away with saying "You'd get optimum area by bending the two planks into a circle."
That reminds me of this old joke:
http://en.wikipedia.org/wiki/Spherical_cow

I didn't say it would fit in a 20 million by 20 million plot, I was saying that it would be very inefficient in terms of area, whereas a square that is 20 by 20 would obviously have plenty of area.

Thanks, guys.
Karen, I haven't built it yet, so it was a question that I just pondered in my wee little brain. That does make sense though that the widths of the boards at the adjoining edges would eat up some of the space. However, even on paper with a pencil thin line it does this space time morphing thing.
Still perplexed.
Just a thought.
PS Good to be back
Yes you are right and it does still do that. I see the new posts have explained it some, but does seem to make better since cause I was still coming up some short also.
Glad you are glad to be back! Have fun!

Could you not make a hexagonal garden?

I didn't say it would fit in a 20 million by 20 million plot, I was saying that it would be very inefficient in terms of area, whereas a square that is 20 by 20 would obviously have plenty of area.
Sorry, but I just couldn't see any other significance in your mentioning a 20 million x 20 million ft garden.

I was demonstrating that no matter how long a rectangle is it will still have less area compared to a square of the same perimeter.

Thanks. I'm starting to get it. My wife said the same thing that LeeE said about my confusing perimeter with area. I think that may be the case. I'm not going to tell her though. She'll make fun of me, do a little "Itoldyouso" dance and her head will swell up. Then we'll have a "Howmuchvolumeismywife'shead?" math question. The problem is getting a little more clear. Still not there yet. The hurt in my brain is becoming more faint. More of a tickle now.
Think I'll go back to the lumber yard for alot more boards. I think the boards to make the 20,000,000 x 20,000,000 will take a few trips. However, I may need that space for all the spherical cows to graze.
Just a thought.

How does the surface area change if the lengths of the boards do not. They are only cut differently. Please help us out.
Just a thought.
Every time you use an angle other than 90 degree, you end up wasting space. Let’s go gaga and use the intense; Triangles:
If you cut one 8 feet board in 3 pieces, you can build a Triangle 2.66666 x 2.66666 x 2.66666; that gives you an area of 3.5555 Sqft.
Now do the same with the other board, and merge the two triangles together, so they form a square. Your square will be 2.66666 x 2.66666 = 7.1111 Sqft.
As you look at your masterwork, you can see the waste caused by building in angles other than 90 degree: 2 wasted boards, diagonally through the square, 5.3333 ft. of total length.
As a side note, that is the reason the base of containers are always round, or square with 90 degree edges.

"Every time you use an angle other than 90 degree, you end up wasting space."
I doubt that.
I'm pretty sure that, the same perimeter more sides is better.
For an octagon you get 19.3 square feet, better than the square, but not as good as the circle.
Also, I'd like to know what you mean by merging two equilateral triangles together to get a square.

You are right! I intended to say every time you use an angle lesser than 90 degree. The merger of the triangles was proposed to face them together to make them assemble a rectangular.