I NEED HELP IN THIS PROBLEM . THE AREA OF A CIRCLE Let r be a positive real numb

Question

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.

Explanation / Answer

for triangle OPQ distance of point Q to projection of Q on line OP is rsin(delta phi) so area of traingle OPQ= 1/2 *base* height = 1/2 * rsin(delta phi) * r = r^2 sin(delta phi ) /2 similarly for triangle OPQ' distance of point Q' to projection of Q' on line OP is rtan(delta phi)area of traingle OPQ'= 1/2 *base* height = 1/2 * rtan(delta phi) * r = r^2 tan(delta phi ) /2 2) since the OQ' distance is large so so the area of the circle considering OQ' will be large , equallity will hold Q' approaches to Q 3) N= 2pie / delta phi

Related Questions

Navigate

• Browse (All)
• Browse I
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.