This Is Not An Exam Question And I Think I Know The Answer!!

[kuiperbelt]

A rope 120 metres long can be formed into any shape so long as the 2 ends are touching. Find the maximum area that can be enclosed by the shape.

[Atheist]

Two and pi and r come into this.

[youngmafbog]

The area will be the same regardless of the shape. so take the easiest shape to work with as being a circle with a circumference of 120M, from this apply the formula for the circumference of a circle which will give you the radius/diameter. Then apply that to the formula for the area of a circle.

You should either know or be able to find out the formula for the area/circumference of a circle.

[ToraToraTora]

I make it 1146 square Metres.

[ClareTG0ld]

"The area will be the same regardless of the shape."

This isn't remotely true. Take a rectangular shape of sides 59.9 metres by 0.1m (area 5.9m^2) and compare to a square shape of sides 30m (area 900m^2).

This is a famous optimisation problem -- for a fixed perimeter, the maximum area that can be bounded is a circle.

[perseverer]

I agree with TTT, not something which happens often, and disagree with YMB's first sentence as it stands.w

[ToraToraTora]

....which is why tinned food is usually in a cylinder!

[douglas9401]

Ye Old Oak ham
Sardines
Fray Bentos pies
Corned beef
Sundry biscuits...

Pseudo cylinders?

[ToraToraTora]

I never said there were none.They are the odd ones out usually supporting a brand. Are you taking on the contrarian mantle doug? By number the vast majority of tinned food is in cylinders because it saves steel.

[douglas9401]

I only post as an intro for your buzzwords dear. Be happy. x

[Hymie]

If the only reason for cylindrical food tins is to minimise the amount of steel used, then the aspect ratio (diameter to length) would similarly be optimised – but this is not the case.

The optimum aspect ratio results in the type of tin that is seen containing fish (somewhat squat/flat); if this was to be applied to larger tins the resultant diameter would be difficult to hold in one hand.

So most food tins have a diameter determined by ergonomics related to humans handling the tin, rather than the minimum amount of metal required to make the tin.

[douglas9401]

And that's why you find catering size cans of veg at the foot of beanstalks.

[mibn2cweus]

A cylinder with the least area to enclose a given volume has a height equal to its diameter. Seen from the side, it looks like a square.

[Hymie]

Of course, a tin made to contain the maximum volume with a minimum amount of material would use a sphere – imagine that on supermarket shelves and in your kitchen cupboard.

[Peter Pedant]

yeah solve 2-pi-arr =120

Proof? I bet Clare has a better one
1965 -WE were told the proof was thus: (*)
It has to be symmetrical ( not sure why) and so square wd be OK but not rectangle. - ( actually you can jump to circle by that alone)

and it is a polygon, with no re-entrant bit zit, U shape- if there is, short cirucit and - draw line across and enlarge area thereby

and the polygons can be made smaller side and larger area - and in the end you get a circle.

we all went ugggh

this is a topological proof which obviously doesnt appeal to 14 y olds

(*) Proof what den? Clare hasnt met the whizzy one liner AB put-down yet. Proof that the largest area is a cirlce
other than the proof: well it just is.

[ClareTG0ld]

I'm not proving this. Not on AB, at least, the syntax I'd need isn't supported. Also, as posed, I think you only need to *know* what the max area shape should be in this situation, rather than set up the complete problem and consider all possible shapes, which I think goes far beyond the realms of A level techniques.

[Peter Pedant]

A cylinder with the least area.....
No - or yes - that is the gold-plated baked bean can problem.
1965 again. - gold plate to minimise the use of gold
can be reduced to one independent variable - suitable for school calculus

otherwise partial differentials are used ( V as a min for both r and h).

The maff master told us the element of a volume of revolution was pi-y2- delta x on the last day of one term
and asked us on the first day of the next.... I recalled pi y2 dee-x
and he went christ. Notice school boy error of delta x confused with dee-x - - oh dear dear ! I got addl-maff O level when I was 14

[Peter Pedant]

Hi Clare
thanks

[New Judge]

//... imagine that on supermarket shelves and in your kitchen cupboard.//

Not to mention how you would get the lid off! :-)

[ZebuSanctuary]

// contain the maximum volume with a minimum amount of material would use a sphere – imagine that on supermarket shelves //

https://bgr.com/science/ooho-water-balls/