Maximum Volume
A sector with central angle is cut from a circle of radius 12 inches (see figure), and the edges of the sector are brought together to form a cone. Find the magnitude of such that the volume of the cone is a maximum.
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Verifying a Formula
(a) Given a circular sector with radius L and central angle (see figure), show that the area of the sector is given by .
(b) By joining the straight-line edges of the sector in part (a), a right circular cone is formed (see figure) and the lateral surface area of the cone is the same as the area of the sector. Show that the area is , where r is the radius of the base of the cone. (Hint: The arc length of the sector equals the circumference of the base of the cone.)
(c) Use the result of part (b) to verify that the formula for the lateral surface area of the frustum of a cone with slant height L and radii r1 and r2 (see figure) is .(Note: This formula was used to develop the integral for finding the surface area of a surface of revolution.)
A cone-shaped drinking cap is made from a circular piece of pater of radius
R by cutting out a sector and joining the edges CA and CB.
Find the maximum capacity of such a cup.
The volume of a frustum of a cone is given by the formula , where x is the radius of the smaller circle, y is the radius of the larger circle and z is the height of the frustum. Find the rate of change of the volume of this frustum when x=10 inches, y=12 inches, and z=18 inches.
