Question

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R/(3)^1/2. Also find the maximum volume.

Answer 1

HELLO DEAR,

let r be the radius and h the height of the inscribed cylinder ABCD.
let V be its volume.
then , V = πr²h------( 2 )
clearly, AC = 2R.
ALSO, AC² = AB² + BC²

=> (2R)² = (2r)² + h²
=> r² = 1/4(4R² - h²)--------( 2 )

using----( 1 ) & -----( 2 )
V = πh/4(4R² - h²)
=> dV/dh = (πR² - 3/4πh²)

and d²V/dh² = -3/2πh.

for Maxima or minima,
we have (dV/dh) = 0

now, dV/dh = 0
=> πR² - 3/4πh² = 0
h = 2R/√3.

therefore,
[d²V/dh²](at h = 2R/√3) = -3/2π * 2R/√3
= -πR√3 < 0.

So, V is maximum when h = 2R/√3.
hence,
the height of the cylinder of maximum volume is 2R/√3.
Largest volume of the cylinder = π×1/4[4R² - 4R²/3] × 2R/√3 = 4πR³/3√3.

I HOPE ITS HELP YOU DEAR,
THANKS