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HSC 2015 MX2 Marathon (archive) (3 Viewers)

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dan964

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Re: HSC 2015 4U Marathon

correct method, yes

(I) is fine. modulus is , argument is which I know you got based by the '...'
(II) is fine as well. As mentioned above by others, modulus is not needed.

tip: except in probability, 'and' usually implies '+'
 
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dan964

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Re: HSC 2015 4U Marathon

Step 1: To produce a formula.
Lets say each had a perimeter of X. X>0. Therefore each side is X/n, n>2 n is an integer

(A) and (B) can be proven by inscribing each REGULAR polygon inside a circle

(A) Each side subtends an angle of 2pi/n at the centre, forming an isosceles triangle
(B) Producing perpendiculars at each side bisects each side, producing sides of X/(2n) and a centre angle of (pi/n)

Using trig. to find the perpendicular gives h=cot(pi/n)*X/2n

Each triangle is therefore has an area of X^2/4n^2*cot(pi/n)

There are n triangles: A = (X^2/4n)*cot(pi/n) is the area (*)

Step 2: Consider the limiting case

The limit of y/tan(y) as y approaches 0 is 1.

Let y be pi/n
cot(pi/n) *pi/n as pi/n approaches 0 is 1.

Therefore as n approaches infinity: cot(pi/n)*1/n equals 1/pi

Side Note: From (*)
Let n approach infinity (the limiting case, i.e. a circle) gives X^2/4pi (by substituting 2pi * r you get the formula)

Differentiate (*) to get
X^2/4n^3 * cosec^2(pi/n) - X^2/2n^3 * cot(pi/n)

X^2/4n^3 * (D+1)^2 where D is cot(pi/n)

which is positive for n>2.
Since the area is positive for n=3, and the dA/dn is positive, as n increases, so does A. So therefore when n is a max, A is a max. Therefore at n --> infinity, A approaches a maximum. (X^2/4pi)
etc.

Therefore circle has the largest area if PERIMETER is constant for all regular polygon.
 
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braintic

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Re: HSC 2015 4U Marathon

tip: except in probability, 'and' usually implies '+'
I couldn't find what this was referring to. Would you explain what you mean in the non-probabilistic sense, perhaps with an example or two.
 

dan964

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Re: HSC 2015 4U Marathon

I couldn't find what this was referring to.
It was a general note, in response to an earlier usage (of which was straightforward)
I understood what he meant, but felt like writing a note.

Would you explain what you mean in the non-probabilistic sense, perhaps with an example or two.
3 and 4 give 7 not 12.
https://www.google.com.au/#q=3+and+4+equals

or the classic joke of 9 and 10 gives 21 (actually 19 but whatever)
 
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Sy123

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Re: HSC 2015 4U Marathon

Step 1: To produce a formula.
Lets say each had a perimeter of X. X>0. Therefore each side is X/n, n>2 n is an integer

(A) and (B) can be proven by inscribing each REGULAR polygon inside a circle

(A) Each side subtends an angle of 2pi/n at the centre, forming an isosceles triangle
(B) Producing perpendiculars at each side bisects each side, producing sides of X/(2n) and a centre angle of (pi/n)

Using trig. to find the perpendicular gives h=cot(pi/n)*X/2n

Each triangle is therefore has an area of X^2/4n^2*cot(pi/n)

There are n triangles: A = (X^2/4n)*cot(pi/n) is the area (*)

Step 2: Consider the limiting case

The limit of y/tan(y) as y approaches 0 is 1.

Let y be pi/n
cot(pi/n) *pi/n as pi/n approaches 0 is 1.

Therefore as n approaches infinity: cot(pi/n)*1/n equals 1/pi

Side Note: From (*)
Let n approach infinity (the limiting case, i.e. a circle) gives X^2/4pi (by substituting 2pi * r you get the formula)

Differentiate (*) to get
X^2/4n^3 * cosec^2(pi/n) - X^2/2n^3 * cot(pi/n)

X^2/4n^3 * (D+1)^2 where D is cot(pi/n)

which is positive for n>2.

Since the area is positive for n=3, and the dA/dn is positive, as n increases, so does A. So therefore when n is a max, A is a max. Therefore at n --> infinity, A approaches a maximum. (X^2/4pi)
etc.

Therefore circle has the largest area if PERIMETER is constant for all regular polygon.
Yes well done, for the other users who attempted this question (your attempt was deleted in a site crash yesterday), the key part I was looking for is what I bolded
 

Drsoccerball

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Re: HSC 2015 4U Marathon

does circle geometry work for conics? I know it says circle but surely conics have some rules as well?
 

braintic

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Re: HSC 2015 4U Marathon

does circle geometry work for conics? I know it says circle but surely conics have some rules as well?
Those 'rules' .... they are all those things you've derived in doing all those Conics questions.
 

Drsoccerball

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Re: HSC 2015 4U Marathon

Those 'rules' .... they are all those things you've derived in doing all those Conics questions.
Oh that makes sense ahahahahahahh can you write the formulas that are assumed knowledge ?
 

Sy123

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Re: HSC 2015 4U Marathon





------------------------------------------------

 
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Sy123

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Re: HSC 2015 4U Marathon

Yep that is correct, your explanation for why those extra terms cancel out is a little vague, but that's the general idea
 

Ekman

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Re: HSC 2015 4U Marathon

Yep that is correct, your explanation for why those extra terms cancel out is a little vague, but that's the general idea
I would of drew an Argand diagram to demonstrate the cancelling out process.
 

Sy123

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Re: HSC 2015 4U Marathon

I would of drew an Argand diagram to demonstrate the cancelling out process.
Right, a more rigorous way of showing it would be to do this:

 
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