Let's break this down and solve it step by step.

Given:

• A closed cylinder is placed inside a sphere such that it just fits.
• The ratio of the height (h) of the cylinder to its diameter (d) is 2:1, so $h = 2d$.
• We need to find the ratio of the volume of the sphere to the volume of the cylinder.

Step 1: Relationship between the dimensions of the cylinder and sphere

Since the cylinder just fits inside the sphere, the diameter of the sphere must be equal to the diagonal of the cylinder. The diagonal of the cylinder can be found using the Pythagorean theorem, as the diagonal spans across the height and the diameter of the cylinder.

The diagonal, $D$, of the cylinder is given by: $D = \sqrt{h^2 + d^2}$ Since $h = 2d$, we can substitute this into the equation: $D = \sqrt{(2d)^2 + d^2} = \sqrt{4d^2 + d^2} = \sqrt{5d^2} = d\sqrt{5}$ Thus, the diameter of the sphere is $D = d\sqrt{5}$, which means the radius $R$ of the sphere is: $R = \frac{d\sqrt{5}}{2}$

Step 2: Volume of the cylinder

The volume $V_{\text{cylinder}}$ of a cylinder is given by: $V_{\text{cylinder}} = \pi r^2 h$ The radius of the cylinder is $r = \frac{d}{2}$, and the height $h = 2d$, so: $V_{\text{cylinder}} = \pi \left(\frac{d}{2}\right)^2 (2d) = \pi \frac{d^2}{4} \times 2d = \frac{\pi d^3}{2}$

Step 3: Volume of the sphere

The volume $V_{\text{sphere}}$ of a sphere is given by: $V_{\text{sphere}} = \frac{4}{3} \pi R^3$ We already know that $R = \frac{d\sqrt{5}}{2}$, so: $V_{\text{sphere}} = \frac{4}{3} \pi \left(\frac{d\sqrt{5}}{2}\right)^3 = \frac{4}{3} \pi \frac{d^3 (5^{3/2})}{8} = \frac{4}{3} \pi \frac{d^3 5\sqrt{5}}{8} = \frac{\pi d^3 5\sqrt{5}}{6}$

Step 4: Ratio of the volumes

Now, we calculate the ratio of the volume of the sphere to the volume of the cylinder: $\text{Ratio} = \frac{V_{\text{sphere}}}{V_{\text{cylinder}}} = \frac{\frac{\pi d^3 5\sqrt{5}}{6}}{\frac{\pi d^3}{2}} = \frac{5\sqrt{5}}{6} \times \frac{2}{1} = \frac{5\sqrt{5}}{3}$

Thus, the ratio of the volume of the sphere to the volume of the cylinder is $\frac{5\sqrt{5}}{3}$.

The correct answer is option (e).

Would you like any further details or have any questions? Here are some related questions you might find interesting:

1. What would be the ratio if the cylinder's height was equal to its diameter?
2. How does the volume of a cone compare to a cylinder with the same height and base?
3. What is the relationship between surface area and volume for a sphere?
4. How does the diagonal of a cuboid change when one dimension is doubled?
5. Can a cylinder fit inside a cube with its height equal to the cube's edge length?

Tip: When calculating volumes for 3D shapes like spheres and cylinders, be mindful of how their dimensions (height, radius, diameter) interrelate!