Find angle theta...please help?

Question

please help i need this...

in terms of theta i need to find the angle that gives me the maximum volume of the cone.
Please without using calculus concepts

you can see the illustration "picture" at
http://img265.imageshack.us/my.php?image=project3yn4.png

"how much of of an angle should i cut of to get the maximum volume of the cone"

Answer 1

Let
R = radius circle
φ = central angle cut out of circle
r = base radius of cone
h = height cone
V = volume cone

The slant height of the cone is the radius of the circle. The slant height of the cone, height of the cone, and base radius of the cone form a right triangle.

h² = R² - r²
h = √(R² - r²)

r = R[(2π - φ)/(2π)]
r = R[1 - φ/(2π)]

The volume of the cone is:

V = (1/3)πr²h
V = (1/3)πr²√(R² - r²)

V = (π/3) R²[(2π - φ)/(2π)]² √(R² - R²[1 - φ/(2π)]²)
V = (πR³/3) (1/4π²) (2π - φ)² √{1 - [1 - φ/π + φ²/(4π²)]}

V = [R³/(12π)] (2π - φ)² √[φ/π - φ²/(4π²)]
V = [R³/(12π)] (2π - φ)² √[(4πφ - φ²)/(4π²)]

V = [R³/(12π)] (2π - φ)² [√(4πφ - φ²)] / (2π)
V = [R³/(24π²)] (2π - φ)² √(4πφ - φ²)

I don't see how to take it beyond this point without calculus.