maximun volume of a cone?

Question

I have a circular piece of paper. And I cut out a sector and connect the two radial edges together to make an open cone. What should the angle of the sector be in order to maximize the volume of the cone?

Answer 1

if a= the angle, then let b=360-a
also let R be the radius of the paper
and r the radius of the cone
and h height of the cone
then r=b/360
and h=sqrt(R^2-r^2)
plug them in the volume formula
v=1/3 pi r^2 h
then you have v as a function of b
then make dv/db=0 for max
the answer will be b=293.76
and a=66.24 degrees.
you have to cut a piece with an angle
of about 66 degrees to get the max volume.

Answer 2

Volume of cone is 1/3 pi r^2 h
where r is the base radius and h is the height.

The angle of the sector will determine how much of the circle's circumference is used to make the circumference of the base of the cone. You can use that to get r (radius of base of cone) in terms of the angle.

You then need a similar expression for h, the height of the cone. It will be determined by the radius of the original circle and the sector angle. Edit: actually, h will be determined more directly by the radius of the original circle and r (which you have just worked out). Simple trig on the right-angled triangle formed by taking a vertical cross-section through the cone.

You can then combine these into the cone volume formula to get an equation in terms of the circle radius (constant) and the sector angle (variable). Use standard differentiation techniques to find its maximum.

Answer 3

Let
R = radius of circular piece of paper
r = base radius of cone
h = height of cone
θ = angle of cutout section of paper

Find θ for maximum volume of cone.
____________

We have:

r = R(2π - θ)/2π

h = √(R² - r²)

V = (1/3)πr²h = (1/3)πr²√(R² - r²)

Take the derivative and set equal to zero to find critical value.

dV/dv = 2πr√(R² - r²) / 3 + (πr²)(1/2)(-2r) / [3√(R² - r²)] = 0
2πr√(R² - r²) / 3 = (πr³) / [3√(R² - r²)]
2√(R² - r²) = r² / √(R² - r²)
2(R² - r²) = r²
2R² - 2r² = r²
3r² = 2R²
r² = (2/3)R²
r = R√(2/3)

Set the two formulas for r equal.

r = R(2π - θ)/2π = R√(2/3)
(2π - θ)/2π = √(2/3)
1 - θ/2π = √(2/3)
θ/2π = 1 - √(2/3)
θ = 2π[1 - √(2/3)] ≈ 1.152986 radians ≈ 66.061231°