Question

A cone-shaped drinking cup is made from a circular piece of paper of radius R by cutting out a sector and joining the edges CA and CB. Find the maximum capacity of such a cup.

Answer 1

Answer:

The maximum capacity of such cup is

2 × 3.142R^3/9×squareroot3

Explanation:

From the diagram in the attachment,

h= height of the cone, r= radius of the cone,R= radius of the original circular piece of paper

Using pythagorean theorem, we get

r^2+h^2=R^2 => r^2=R^2-h^2

Formular for the volume of a cone is given as V=1/3×(3.142)^2h

Substituting for r^2 when R is a constant

V=1/3(3.142)(R^2-h^2)h

V=1/3(3.142)(R^2-h^3)

V'=0=1/3(3.142)R^2-3h^2)

0=R^2-3h^2

h^2=R^2/3 => h=R/squareroot of 3

V"=1/3(3.142)(0-6)=-2×3.142×h

Negative shows that v is concave

V(R/squareroot of 3)=1/3(3.142)[R^2×R/squareroot 0f 3-(R/squareroot of 3)^3

V=2×3.142×R^3/9× squareroot of 3