# Math

posted by .

A cone is constructed by cutting a sector from a circular sheet of metal with radius 20cm. The cut sheet is then folded up and welded. Find the radius and height of the cone with the maximum volume that can be formed in this way.

• Math -

Given an arc of x radians, you should be able to convince yourself that you want the maximum volume

v = 1/3 pi r^2 h
where r = x/2pi * 20 = 10x/pi
and h = √(400-r^2)

so,

v = 1/3 pi (10x/pi)^2 √(400-(10x/pi)^2)
= 1/3 pi (10/pi)^3 x^2 √(4pi^2-x^2)

Note that as expected,
v(0) = 0 (zero radius base)
v(2pi) = 0 (zero height)

so, maximum volume must be somewhere in between.

dv/dx = 1000x/3pi^2 (8pi^2 - 3x^2)/√(4pi^2-x^2)
dv/dx = 0 when x=0 (minimum volume)
or x = √(8pi^2/3) (maximum volume)

so, for that value of x,
r = 10/pi √(8pi^2/3)
h = √(400 - 100pi^2 * 8pi^2/3) = 20/√3

max v is thus pi/3 (100/pi^2)(8pi^2/3) (20/√3) = 2000pi/(9√3)

As usual, check my math.

## Similar Questions

1. ### math

Dana takes a sheet of paper, cuts a 120-degree circular sector from it, then rolls it up and tapes the straight edges together to form a cone. Given that the sector radius is 12 cm, find the height and volume of this paper cone.
2. ### geometry

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 …
3. ### Geometry

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 …
4. ### Math

Cone Problem Beginning with a circular piece of paper with a 4- inch radius, as shown in (a), cut out a sector with an arc of length x. Join the two radial edges of the remaining portion of the paper to form a cone with radius r and …
5. ### math

Two right circular cone, one upside down in the other. The two bases are parallel. The vertex of the smaller cone lies at the center of the larger cone’s base. The larger cone’s height and base radius are 12 and 16 ft, respectively. …
6. ### calculus

A paper cone is to be formed by starting with a disk of radius 9cm, cutting out a circular sector, and gluing the new edges together. The size of the circular sector is chosen to maximize the volume of the resulting cone. How tall …
7. ### calculus

A paper cone is to be formed by starting with a disk of radius 9cm, cutting out a circular sector, and gluing the new edges together. The size of the circular sector is chosen to maximize the volume of the resulting cone. How tall …
8. ### calculus

A paper cone is to be formed by starting with a disk of radius 9cm, cutting out a circular sector, and gluing the new edges together. The size of the circular sector is chosen to maximize the volume of the resulting cone. How tall …
9. ### Calculus

A cone is inscribed in a sphere of radius a, centred at the origin. The height of the cone is x and the radius of the base of the cone is r, as shown in the diagram opposite. Find the height, x, for which the volume of the cone is …
10. ### Calculus

A funnel is constructed by removing a sector form a circular metal sheet with 7 inch radius. Determine the maximum volume of the funnel constructed in this way if the small amount of volume lost at the tip of the funnel is neglected.

More Similar Questions