A cylinder is inscribed in a ball of radius r. the diagonal of the axial section of the cylinder is inclined

A cylinder is inscribed in a ball of radius r. the diagonal of the axial section of the cylinder is inclined to the base at an angle of 30 °. Find the volume of the cylinder.

The diagonal of the axial section of the cylinder d is equal to the diameter of the ball or twice the radius of the ball: d = 2r.

The projection of the diagonal onto the base of the cylinder is equal to the diameter of the cylinder d1, and the projection onto the lateral side is equal to the height of the cylinder h.

d1 / d = cos 30 °;

d1 = d * cos 30 ° = 2r * √3 / 2 = r√3;

h / d = sin 30 °;

h = d * sin 30 ° = 2r * 0.5 = r;

V = (π * d1 ^ 2 * h) / 4 = (π * (r√3) ^ 2 * r) / 4 = 3πr ^ 3/4.

Answer: V = 3πr ^ 3/4.