My Math Forum Area of a Circle problem

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 August 6th, 2012, 08:09 AM #1 Member   Joined: Sep 2007 Posts: 77 Thanks: 0 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!
 August 6th, 2012, 10:15 AM #2 Math Team     Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory Re: Area of a Circle problem It is straightforoward that the x coordinate of Q' is simply r. Now, $Q'P = OP \tan(\theta) = r \tan(\theta)$ So, the y coordinate of Q' is $r \tan(\theta)$ Can you proceed for Q now?
 August 6th, 2012, 10:59 AM #3 Member   Joined: Sep 2007 Posts: 77 Thanks: 0 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?
 August 6th, 2012, 12:42 PM #4 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 466 Math Focus: Calculus/ODEs Re: Area of a Circle problem 10.) i) We may use the definition of sine and cosine to find: $\cos$$\theta$$=\frac{Q_x}{r}\:\therefore\:Q_x=r\co s$$\theta$$$ $\sin$$\theta$$=\frac{Q_y}{r}\:\therefore\:Q_y=r\si n$$\theta$$$ Since $Q_x'=r$ we then find $\tan$$\theta$$=\frac{Q_y'}{r}\:\therefore\:Q_y #39;=r\tan$$\theta$$$ ii) a) Using the formula for the area of a triangle $A=\frac{1}{2}bh$ we then find: $A=\frac{1}{2}r\cdot r\sin$$\Delta\varphi$$=\frac{r^2\sin$$\Delta\varph i$$}{2}$ $A'=\frac{1}{2}r\cdot r\tan$$\Delta\varphi$$=\frac{r^2\tan$$\Delta\varph i$$}{2}$ b) For any natural number $N\ge3$ and $r>0$ 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) $\Delta\varphi=\frac{2\pi}{N}\:\therefore\:N=\frac{ 2\pi}{\Delta\varphi}$ and hence: $C_r=\lim_{\Delta\varphi\to0}\frac{2\pi r^2\sin$$\Delta\varphi$$}{2\Delta\varphi}=\lim_{\D elta\varphi\to0}\frac{\pi r^2\sin$$\Delta\varphi$$}{\Delta\varphi}=$ $\pi r^2\lim_{\Delta\varphi\to0}\frac{\sin$$\Delta\varp hi$$}{\Delta\varphi}=\pi r^2(1)=\pi r^2$
 August 6th, 2012, 12:47 PM #5 Member   Joined: Sep 2007 Posts: 77 Thanks: 0 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!
 August 6th, 2012, 01:02 PM #6 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 466 Math Focus: Calculus/ODEs Re: Area of a Circle problem To be more formal about the last part, we could state: $\pi r^2\lim_{\Delta\varphi\to0}\frac{\sin$$\Delta\varp hi$$}{\Delta\varphi}\le C_r\le\pi r^2\lim_{\Delta\varphi\to0}\frac{\tan$$\Delta\varp hi$$}{\Delta\varphi}$ Apply L'Hpital's rule to the limits (although this isn't necessary, just quicker): $\pi r^2\lim_{\Delta\varphi\to0}\cos$$\Delta\varphi$$\l e C_r\le\pi r^2\lim_{\Delta\varphi\to0}\sec^2$$\Delta\varphi$$$ $\pi r^2\le C_r\le\pi r^2$

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### let r be a positive real number and let o 􏰆 􏰎0, 0􏰐 and p 􏰆 􏰎r, 0􏰐 be points in the coordinate plane. let ℓθ be the (half )-line that emanates from o and makes an angle θ with respect to the positive x-axis (see ﬁgure). let q be

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