Bunuel wrote:

A cube of side s is positioned within a sphere such that each of its corners is tangent to the inside of the sphere and such that the two figures do not otherwise interact. What is the volume of the sphere, in terms of s? (Note: the volume of a sphere is equivalent to \(\frac{4}{3} \pi r^3\).)

A. \(\frac{s^3\pi \sqrt{2}}{3}\)

B. \(2s^3 \pi \sqrt{3}\)

C. \(\frac{8s^3\pi \sqrt{2}}{3}\)

D. \(4s^3 \pi \sqrt{3}\)

E. \(\frac{s^3\pi \sqrt{3}}{2}\)

The cube's corners are tangent to the inside of the sphere.

No other parts of the 3D figures touch.

Thus the cube's space diagonal (the line between two corners that do not share a face) = the diameter of the sphere

The cube's space diagonal, d, is

\(s\sqrt{3}\)OR: find the space diagonal from a variation of Pythagorean theorem*:

\(s^2 + s^2 + s^2 = d^2\)

\(3s^2 = d^2\)

\(\sqrt{3s^2}=\sqrt{d^2}\)

\(d = s\sqrt{3}\)Cube's space diagonal = sphere's diameter, and

\(\frac{diameter}{2}\) = sphere's radius

\(r = \frac{s\sqrt{3}}{2}\)Volume, V, of sphere:

\(\frac{4}{3} \pi r^3\)

\(r = \frac{s\sqrt{3}}{2}\)

\(V=\frac{4}{3} \pi* (\frac{s\sqrt{3}}{2})^3\)

\(V= \frac{4}{3} \pi* \frac{s^3 3\sqrt{3}}{8}\)

\(V=\frac{s^3\pi \sqrt{3}}{2}\)Answer E

Space diagonal of any rectangular prism:

\((l^2 + w^2 + h^2) = d^2\). Square side \(s = l, w,\) and \(h\).
_________________

In the depths of winter, I finally learned

that within me there lay an invincible summer.

-- Albert Camus, "Return to Tipasa"