Rabbit run for kids

I have been helping my 13 year old daughter with her maths and have been setting some exam style questions for her. Here is the latest.

She has rabbits and has purchased 12M of fence to make a run in the garden. Which shape out of and equilateral triangle, square and a circle would provide the most area for her rabbits to play in.

Please please, no Phd maths wizards saying how easy it is. This is for kids but that said I found it interesting the number of adults that I asked who thought that a fixed perimeter would give a fixed area.

Nice question.

For those who _do_ know the answer already, how about this variant:

The run must still be only an equilateral triangle, square or circle, but one side of the run may be formed (where possible) by the side of the house. Which shape is best now?

My house has a concave semicircle indent with a curve radius of ~7.64m, so still the circle...

Originally posted by forkedknight
My house has a concave semicircle indent with a curve radius of ~7.64m, so still the circle...

If you have another concave semicircle indent with a radius of ~5.09m then you might get away with this ...

Is the circle the maximum enclosed area per unit perimeter for all 2D geometric shapes?

Originally posted by joe shmo
Is the circle the maximum enclosed area per unit perimeter for all 2D geometric shapes?

Yes

Originally posted by forkedknight
Yes

Ok, thought so...but I never investigated it. How would you go about proving it?

Basically, for any N, an N+1 sided regular polygon has more area than an N-sided polygon

N -> infinity -> circle

Hi Joe, The way I worked the problem with my daughter was to get her to look up the formulas for areas of shapes. Triangle is half base x perpendicular height, square L x B and circle Pi x r SQ. For the circle to derive the radius you need Circumference = Pi x 2r. With the perimeters fixed at 12M just a bit of algebra with the formulas.

The triangle took her longer than I planned as some pythagorus was needed to find the perpendicular height.

All in all she did well. It was also a good exercise to get her to show all working for maximum marks.

It can also be proven by experiment.
When you blow a soap bubble, the soap film wants to collapse inward, so it tries to contain the air with as small a surface area as possible, what 3d shape results? A similar thing in 2d is an elastic band stretched around a bunch of spaghetti: Does the shape of the band approximate a circle, a triangle or a square?

Originally posted by eltricky
Hi Joe, The way I worked the problem with my daughter was to get her to look up the formulas for areas of shapes. Triangle is half base x perpendicular height, square L x B and circle Pi x r SQ. For the circle to derive the radius you need Circumference = Pi x 2r. With the perimeters fixed at 12M just a bit of algebra with the formulas.

The triangle took ...[text shortened]... all she did well. It was also a good exercise to get her to show all working for maximum marks.

Yeah, sounds like fun! I hope she continues to find an interest in math as she grows up. I didn't catch the math bug till my early 20's, at which point I had to learn from the very basics. ( I couldn't add fractions when I decided to attend college) Now,(in my late 20's) the subject is a very big part of my everyday life. Keep up the good work!

Originally posted by iamatiger
It can also be proven by experiment.
When you blow a soap bubble, the soap film wants to collapse inward, so it tries to contain the air with as small a surface area as possible, what 3d shape results? A similar thing in 2d is an elastic band stretched around a bunch of spaghetti: Does the shape of the band approximate a circle, a triangle or a square?

So, do you believe that the Mathematics is a manifestation of the Physics, vice versa, or niether? That is to say, can a mathematical concept be proven by physical experiment, and the converse also?

Originally posted by joe shmo
So, do you believe that the Mathematics is a manifestation of the Physics, vice versa, or niether? That is to say, can a mathematical concept be proven by physical experiment, and the converse also?

Mathematics is a philosophical discipline. It works with a set of axoims and Operators and expands from there. Some physical phenomena can be described by mathematical formula.

It is pleasing to find a simple mathematical Expression to explain physical behaviour, but it is not proof that the theoretical Approach is correct.
On the other hand minimization Problems occur often in nature. It is not a proof of a mathematical minimization algorithm if you find a physical example (think dense sphere packings).

Hi Jo, I have always believed that the two subjects are intertwined. It surprises me that maths is not taught with more 'real life' scenarios. I find that people, especially kids learn and retain the information if they can see a reason for doing it.

With my daughter I use a wage slip when teaching percentages, pizza slices for fractions.

With physics a garden hose to show pressure=force/area. Adding more hose to show restrictions to flow. Expansion and contraction can be explained with a plastic drinks bottle collapsing in a freezer as the air cools.

She shows far more interest by me teaching her in this manner and enjoys doing the experiments herself instead of everything being bookwork.

By the way, im not a teacher. My trade is electrical engineer. I just work with what I believe to be a basic human trait, that people put in more effort if they are interested in something and can see a reason for doing it.

Originally posted by eltricky
Hi Jo, I have always believed that the two subjects are intertwined. It surprises me that maths is not taught with more 'real life' scenarios. I find that people, especially kids learn and retain the information if they can see a reason for doing it.

With my daughter I use a wage slip when teaching percentages, pizza slices for fractions.

With physi ...[text shortened]... people put in more effort if they are interested in something and can see a reason for doing it.

Yeah, its very good to have an intuitive grasp of physical systems. It makes the application of the mathematics to that system all the easier when its time to quantify some or all aspects of it. I'm a mechanical engineer, so I'm constantly racking my brain in an attempt to join the two (reality and math)...It seems to be a slow, and almost all times, a frustrating process (as I'm sure you can agree)!