posted by Sarah
11. Infinitely many different sectors can be cut from a circular piece of paper with a 12-cm radius, and any such sector can be fashioned into a paper cone with a 12-cm slant height.
(a) Show that the volume of the cone produced by the 180-degree sector is larger than the volume of the cone produced by the 120-degree sector.
(b) Find a sector of the same circle that will produce a cone whose volume is even larger.
(c) Express the volume of a cone formed from this circle as a function of the central angle of the sector used to form it, then find the sector that produces the cone of greatest volume.
This is a question that needs to be visualized.
(I used to go to our teacher's staff room and bring back a cone paper cup from the water cooler and cut it open)
One has to realize that the arclength of the sector becomes the circumference of the circle of the cone, and the original radius of the sector becomes the slant height of the cone.
so for a 180° sector arclength = (1/2)(2π)12 = 12π cm
so the circumference of the base circle of the cone = 12π
2πr = 12π and the radius of the cone base is 6
then the height h is ...
h^2 + 6^2 = 12^2
h = √108 = appr. 10.3923
Volume of cone = (1/3)πr^2h = (1/3)π(36)√108 = 391.78
since 120° is 1/3 of the rotation, the arclength would be (1/3) of 2π(12) or 8π
find the radius of the base circle as above, then the height, and finally the volume.