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Thursday, January 21, 2010

What is the largest possible volume a right circular cylinder can have if it is inscribed in a sphere of radius 5?

Chris said...: Let the radius of the sphere be a, the radius of cylinder be r and the half height of cylinder be h.

This really needs a sketch.
Pythagoras says a² = h² + r²

The volume of the cylinder, V = 2πr²h = 2πr²√(a²-r²)

dV/dr = 4πr√(a²-r²) - 2πr³/√(a²-r²)
V is a max when dV/dr = 0
=> 4πr√(a²-r²) = 2πr³/√(a²-r²)
=> 2(a²-r²) = r²
=> r = a√(2/3)
So Vmax = 2πr²√(a²-r²) = (4/3)πa³/√3
As a = 5, Vmax = 302.3000 (4D); January 21, 2010 9:51 PM
Chris said...: I did that the hard way - so easy way.

Let the radius of the sphere be a, the radius of cylinder be r and the half height of cylinder be h.

This really needs a sketch.
Pythagoras says a² = h² + r² => r² = a² - h²

The volume of the cylinder, V = 2πr²h = 2π(a²-h²)h
= 2πa²h -2πh³
dV/dh = 2πa² -6πh²
Max when dV/h = 0 => h = a/√3

So Vmax = 2π(a²-a²/3)a/√3 = (4/3)πa³/√3

As a = 5, Vmax = 302.3000 (4D); January 21, 2010 10:51 PM
Zaux said...: Hi Chris,
nice work ... also correct; January 22, 2010 5:53 AM