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August 6th, 2012, 08:09 AM   #1
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Area of a Circle problem

The area of a circle
i. Let r be a positive real number and let O = (0, 0) and P = (r, 0) be points in the coordinate plane. Let ?? be the (half)-line that emanates from O and makes an angle ? with respect to the positive x-axis (see figure in attachment).
Let Q be the point on ?? whose distance to O is r and let Q' be the point on ?? whose perpendicular projection onto the positive x-axis is P (see figure in attachment).
Find the x and y coordinates of Q and Q'.

ii. Partition the angle 2? into N equal parts of size ? and let
Cr = area of a circle with radius r,
A = area of the triangle ?OPQ,
A' = area of the triangle ?OPQ',
where O,P,Q,Q' are as in the above figure with angle ? = ?.

(a) Show that
A = r^2 sin(?) / 2
and
A' = r^2 tan(?) / 2

(b) Explain why the inequalities
Nr2 sin(?) / 2 ? Cr ? Nr2 tan(?)/ 2

hold for any r > 0 and any natural number N ? 3.

(c) Find an expression for N in terms of ? and take the limit as ? --> 0 in (7)
to find Cr.


Can anyone help get me started on this? Thank you!
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August 6th, 2012, 10:15 AM   #2
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Re: Area of a Circle problem

It is straightforoward that the x coordinate of Q' is simply r.

Now,

So, the y coordinate of Q' is

Can you proceed for Q now?
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August 6th, 2012, 10:59 AM   #3
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Re: Area of a Circle problem

From there, I know that the coordinates of QP are (rcos x, rsinx). To find the area, I need to take the anti-derivative of the the function. However, even though we have the two coordinates, how do I know which formula to take the anti-derivative of?
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August 6th, 2012, 12:42 PM   #4
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Re: Area of a Circle problem

10.)

i) We may use the definition of sine and cosine to find:





Since we then find

ii)

a) Using the formula for the area of a triangle we then find:





b) For any natural number and the geometrical interpretation of the given inequality is that for a circle of radius r, its area is greater than an inscribed N-gon and less than that of a circumscribed N-gon.

c) and hence:



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August 6th, 2012, 12:47 PM   #5
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Re: Area of a Circle problem

Thank you for the help! As you can see from my prior post, I was really off from what I needed to do, so I appreciate the help in preparation for my final. Thank you thank you thank you!
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August 6th, 2012, 01:02 PM   #6
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Re: Area of a Circle problem

To be more formal about the last part, we could state:



Apply L'Hpital's rule to the limits (although this isn't necessary, just quicker):



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