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6 Nov 2019
I NEED HELP IN THIS PROBLEM .
THE AREA OF A CIRCLE 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 theta with respect to the positive x-axis (see figure). 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 4). Find the x and y coordinates of Q and Q'. Partition the angle 2pi into N equal parts of size Delta phi and let Cr = area of a circle with radius r, A = area of the triangle Delta OPQ, A' = area of the triangle Delta OPQ', where O, P, Q, Q' are as in the above figure with angle theta = Delta phi. Figure 4: The geometric construction from problem 10. Show that A = r2 sin (Delta phi)/2 and A' = r2 tan(Delta phi)/2. Explain why the inequalities Nr2 sin (Delta phi)/2 Cr Nr2 tan (Delta phi)/2. hold for any r > 0 and any natural number N 5. Find an expression for N in terms of Delta phi and take the limit as Delta phi rightarrow 0 in (7) to find Cr. Show transcribed image text
I NEED HELP IN THIS PROBLEM .
THE AREA OF A CIRCLE 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 theta with respect to the positive x-axis (see figure). 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 4). Find the x and y coordinates of Q and Q'. Partition the angle 2pi into N equal parts of size Delta phi and let Cr = area of a circle with radius r, A = area of the triangle Delta OPQ, A' = area of the triangle Delta OPQ', where O, P, Q, Q' are as in the above figure with angle theta = Delta phi. Figure 4: The geometric construction from problem 10. Show that A = r2 sin (Delta phi)/2 and A' = r2 tan(Delta phi)/2. Explain why the inequalities Nr2 sin (Delta phi)/2 Cr Nr2 tan (Delta phi)/2. hold for any r > 0 and any natural number N 5. Find an expression for N in terms of Delta phi and take the limit as Delta phi rightarrow 0 in (7) to find Cr.
Show transcribed image text Beverley SmithLv2
6 Oct 2019