Paper Cone
Paper Cone
A cone-shaped paper drinking cup is to hold 90 cubic cm of water.?
A cone-shaped paper drinking cup is to hold 90 cubic cm of water. Find the height and radius of the cup that will require the least amount of paper.
Make sure your answer is in terms of Pi.
Let:
R be the radius of the paper,
a be the angle removed,
r be the radius of the cone,
h be the height of the cone,
v be the volume of the cone,
t be the semi-vertical angle of the cone.
The circumference of the top of the cone is:
2pi r.
The length of the curved edge of the paper disc with the sector removed is:
(2pi - a)r.
Equating these two values:
2pi r = (2pi - a)R
r / R = 1 - a / 2pi ...(1)
The volume of the cone is:
v = (pi / 3)r^2 h ...(2)
From the geometry of the cone:
r^2 = R^2 - h^2 ...(3)
Substituting in (2) for r from (3):
v = (pi / 3) h(R^2 - h^2)
= (pi / 3)(R^2 h - h^3)
Differentiating with respect to h:
dv / dh = (pi / 3)(R^2 - 3h^2)
d^2v / dh^2 = - 2pi h
dv / dh = 0 when h^2 = R^2 / 3
and d^2v / dh^2 is clearly negative, indicating a maximum.
Substituting this value of h in (3):
(r / R)^2 = 2 / 3 ...(4)
Substituting for r / R in (1):
1 - a / 2pi = sqrt(2 / 3)
a = 2pi(1 - sqrt(2 / 3)).
When the volume v is maximum, the triangle formed by h, r and R gives:
sin(t) = sqrt(2 / 3)
cos(t) = sqrt(1 / 3)
tan(t) = sqrt(2)
Therefore:
v = pi r^2 h / 3
= pi h^3 tan^2(t) / 3
h = [ 3v / (2pi) ]^(1 / 3).
v = pi r^3 / [ 3 tan(t) ]
= pi r^3 / [ 3 sqrt(2) ]
r = [ 3v sqrt(2) / pi ]^(1 / 3).
Putting v = 90 cm^3 gives:
h = [ 135 / pi ]^(1 / 3) cm.
r = [ 270 sqrt(2) / pi ]^(1 / 3) cm.
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Tip #4: Making a Paper Cone (Cornet)
