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- - **Area of a Circle problem**
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Area of a Circle problemThe 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 letC r = 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 C r.Can anyone help get me started on this? Thank you! |

Re: Area of a Circle problemIt 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? |

Re: Area of a Circle problemFrom 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? |

Re: Area of a Circle problem10.) 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: |

Re: Area of a Circle problemThank 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! |

Re: Area of a Circle problemTo be more formal about the last part, we could state: Apply L'Hôpital's rule to the limits (although this isn't necessary, just quicker): |

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