Çözüldü Pythagorean Theorem - Cone and Sphere Volumes

A question translated from Turkish and checked and slightly revised by the AI at www.mathful.com:

What is the ratio of the volume of the cone of maximum volume inscribed in a sphere, with its base being the circular cross-section formed by a plane cutting the sphere at a distance equal to one-fourth of the sphere's diameter from the center, to the volume of the sphere?

A) 5 / 16
B) 9 / 32
C) 1 / 4
D) 7 / 26
E) 5 / 16

Sphere's Radius: r
Cone's Radius: R
Taking the cross section into account and considering the right triangle whose two edges and hypotenuse are 2r / 4, R, and r, respectively, then applying Pythagorean Theorem;
r^2 = (r / 2)^2 + R^2 ⇒ R^2 = 3r^2 / 4 ⇒ V(cone) = (1 / 3)·π·(3r^2 / 4)·(r + r / 2) = (3π / 8)·r^3....(I)
V(sphere) = (4π / 3)·r^3....(II)
Dividing (I) into (II) and simplifying yields (3 / 8)(3 / 4) = 9 / 32.

Source: "Üniversiteye Hazırlık - Geometri", FDD Yayınları, 2001, Page 139, Question 16