Thursday

August 21, 2014

August 21, 2014

Posted by **Sarah** on Thursday, April 15, 2010 at 12:01pm.

(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.

- geometry -
**Reiny**, Thursday, April 15, 2010 at 2:09pmThis 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

b)

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.

- geometry -
**sarah**, Thursday, April 15, 2010 at 8:41pmthanks

**Related Questions**

Geometry - Infinitely many different sectors can be cut from a circular piece of...

Math - Cone Problem Beginning with a circular piece of paper with a 4- inch ...

PreCalculus - Cone Problem Beginning with a circular piece of paper with a 4- ...

Math - I already posted this, but wanted to say that I have the answers and it ...

CALCULUS - A cone shaped paper drinking cup is to be made from a circular piece ...

math - A circular sector of central angle theta and 9 cm radius can be fashioned...

geometry - A cone is created from a paper circle with a 90° sector cut from it. ...

math - Dana takes a sheet of paper, cuts a 120-degree circular sector from it, ...

calculus - A paper cone is to be formed by starting with a disk of radius 9cm, ...

calculus - A paper cone is to be formed by starting with a disk of radius 9cm, ...