need help with d) 4. In this question, we will determine the optimal dimensions of a circular paper sector used to create aconical filter for brewing coffee. The circular sector (see diagram) has radius r and subtended angle 8,resulting in an arc length of re and an area of A = 21/2²20.r²0. By joining the two radial edges of the sector, weobtain a circular cone whose base has circumference re and a slant height of r (see diagram).0↑OrFigure 2: Schematic for Question 4 showing how a circular sector is turned into a circular cone. (a) If the height of the resulting cone is ŉ and the radius of the cone's base is R, show thatre27R =and(b) Show that the volume of the resulting cone, V =V-h ==FR²,r³8224π²V(r) =r2πV-√4π² - 0².R²h, can be expressed as√4π² - 0².(c) The manufacturers of these paper filters require that the sector's area, A, is a fixed constant. Showthat the resulting cone's volume can be expressed asA³32r A²1(d) Determine the value of the radius r* such that V(r*) is a stationary point.(You do not need to show that r* is a maximum, though this fact is also true).(e) Determine the value of the sector angle 0* that maximises the resulting cone's volume.(Your answer should not depend on A, which shows that the optimal angle is the same regardless ofthe area of the sector!)

Given is a sector of radius r, a subtended angle θ, circumference rθ and area as A=12r2θ. It is…

(a) If the height of the resulting cone is h and the radius of the cone's base is R, show that R = 0|55 re and r h = •√4π² - 0². 27T (b) Show that the volume of the resulting cone, V = r³8² 247² V = R²h, can be expressed as -√477² 0². (c) The manufacturers of these paper filters require that the sector's area, A, is a fixed constant. Show

(a) If the height of the resulting cone is h and the radius of the cone's base is R, show that R = 0|55 re and r h = •√4π² - 0². 27T (b) Show that the volume of the resulting cone, V = r³8² 247² V = R²h, can be expressed as -√477² 0². (c) The manufacturers of these paper filters require that the sector's area, A, is a fixed constant. Show

Elementary Geometry for College Students

6th Edition

ISBN:9781285195698

Author:Daniel C. Alexander, Geralyn M. Koeberlein

Publisher:Daniel C. Alexander, Geralyn M. Koeberlein

Chapter6: Circles

Section6.3: Line And Segment Relationships In The Circle

Problem 39E: The center of a circle of radius 2 inches is at a distance of 10 inches from the center of a circle...

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