A sector with central angle theta is cut from a circle of radius 11 inches, and

Question

A sector with central angle theta is cut from a circle of radius 11 inches, and the edges of the sector are brought together to form a cone. Find the magnitude of theta such that th volume of the cone is a maximum.

Explanation / Answer

The perimeter of a sector with a central angle of ? and a radius of r is: s = r?. Since r = 13: s = 13?. Thus, the circumference of the base of the cone is 13?. Then, the radius of the base is: 2pr = 11? ==> r = 11?/(2p). If l is the slant height of the cone, then: l = v(r^2 + h^2) ==> h = v(l^2 - r^2), where r is the radius of the base = v{11^2 - [13?/(2p)]^2} = v[121 - 169?^2/(4p^2)] = v[(676p^2 - 169?^2)/(4p^2)]. The volume of the cone in terms of ? is then: V = (1/3)pr^2*h = (1/3)p[11?/(2p)]^2v[(676p^2 - 169?^2)/(4p^2)]. by using calculator it is maximized when when ? = 1/11 pi radians

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