Question

Find the dimensions and volume of the right circular cylinder of maximum volume inscribed in a sphere with radius 35cm

Answer 1

Let
R = radius of sphere
r = radius of cylinder
h = height of cylinder
V = volume cylinder

Answer 2

V = πr²h
r² = V/(πh)

By the Pythagorean Theorem

R² = r² + (h/2)² = r² + h²/4
r² = R² - h²/4
r = √(R² - h²/4)

Answer 3

Set the equations for r² equal.

V/(πh) = R² - h²/4
V = πR²h - πh³/4

Take the derivative with respect to h and set it equal to zero to find the critical point(s).

dV/dh = πR² - 3πh²/4 = 0
πR² = 3πh²/4
4R²/3 = h²
h = 2R/√3

Answer 4

Take the second derivative to determine the nature of the critical points.

d²V/dh² = -2*3πh/4 = -3πh/2 < 0 implies relative maximum

Solve for r.

r = √(R² - h²/4)
r = √{R² - (2R/√3)²/4} = √{R² - R²/3}
r = √{(2/3)R²} = R√(2/3)

Compute the volume of the cylinder.

V = πr²h = π{R√(2/3)}²{2R/√3} = π{(2/3)R²}{2R/√3}
V = 4πR³/(3√3) = (4√3)πR³/9

Answer 5

Thank you :)