correct method, yes
Step 1: To produce a formula.
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.tip: except in probability, 'and' usually implies '+'
It was a general note, in response to an earlier usage (of which was straightforward)I couldn't find what this was referring to.
3 and 4 give 7 not 12.Would you explain what you mean in the non-probabilistic sense, perhaps with an example or two.
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 boldedStep 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.
Those 'rules' .... they are all those things you've derived in doing all those Conics questions.does circle geometry work for conics? I know it says circle but surely conics have some rules as well?
Oh that makes sense ahahahahahahh can you write the formulas that are assumed knowledge ?Those 'rules' .... they are all those things you've derived in doing all those Conics questions.
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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.Yep that is correct, your explanation for why those extra terms cancel out is a little vague, but that's the general idea
Right, a more rigorous way of showing it would be to do this:I would of drew an Argand diagram to demonstrate the cancelling out process.