We are tasked with determining the value of $k$, the ratio of the volume of the original cylinder to that of the smaller cylinder, in terms of the central angle $\theta$ subtended by the radius of the smaller cylinder.

Step 1: Original setup

We have:

• A cylinder with radius $r$ and height $h$ inscribed in a sphere with radius $R$.
• The cylinder is cut by a plane parallel to its base, producing a smaller cylinder at the top and a frustum below.

Let the smaller cylinder have a radius $r_1$ and height $h_1$, and the central angle $\theta$ is subtended by the radius $r_1$ of the smaller cylinder at the base of the original cylinder.

Step 2: Geometry of the inscribed cylinder and sphere

The cylinder is inscribed in the sphere, meaning that the diagonal of the cylinder (which is the space diagonal from the center of the top to the center of the bottom) is equal to the diameter of the sphere, $2R$. The height $h$ of the cylinder relates to the radius of the sphere as follows: $h = 2\sqrt{R^2 - r^2}$ where $r$ is the radius of the cylinder.

Step 3: Volume of the original cylinder

The volume $V_{\text{cyl}}$ of the original cylinder is given by: $V_{\text{cyl}} = \pi r^2 h$ Substitute the expression for $h$: $V_{\text{cyl}} = \pi r^2 \cdot 2\sqrt{R^2 - r^2}$ Thus, the volume of the original cylinder is: $V_{\text{cyl}} = 2\pi r^2 \sqrt{R^2 - r^2}$

Step 4: Volume of the smaller cylinder

Now, consider the smaller cylinder. Its radius is $r_1 = r \cos\frac{\theta}{2}$, since the angle $\theta$ subtends the radius $r_1$ at the base of the original cylinder. The height of the smaller cylinder is $h_1$, which corresponds to the height of the slice formed by the plane at height $h_1$.

From the geometry, the height of the smaller cylinder is related to the angle $\theta$ by the vertical projection in the sphere, giving: $h_1 = 2\sqrt{R^2 - r_1^2} = 2\sqrt{R^2 - r^2 \cos^2\frac{\theta}{2}}$ Thus, the volume $V_{\text{small}}$ of the smaller cylinder is: $V_{\text{small}} = \pi r_1^2 h_1 = \pi \left( r \cos\frac{\theta}{2} \right)^2 \cdot 2\sqrt{R^2 - r^2 \cos^2\frac{\theta}{2}}$ Simplifying: $V_{\text{small}} = 2\pi r^2 \cos^2\frac{\theta}{2} \cdot \sqrt{R^2 - r^2 \cos^2\frac{\theta}{2}}$

Step 5: Ratio of volumes

The ratio $k$ of the volume of the original cylinder to that of the smaller cylinder is: $k = \frac{V_{\text{cyl}}}{V_{\text{small}}}$ Substituting the expressions for $V_{\text{cyl}}$ and $V_{\text{small}}$: $k = \frac{2\pi r^2 \sqrt{R^2 - r^2}}{2\pi r^2 \cos^2\frac{\theta}{2} \cdot \sqrt{R^2 - r^2 \cos^2\frac{\theta}{2}}}$ Simplifying further: $k = \frac{\sqrt{R^2 - r^2}}{\cos^2\frac{\theta}{2} \cdot \sqrt{R^2 - r^2 \cos^2\frac{\theta}{2}}}$

Final Answer:

Thus, the value of $k$ is: $k = \frac{\sqrt{R^2 - r^2}}{\cos^2\frac{\theta}{2} \cdot \sqrt{R^2 - r^2 \cos^2\frac{\theta}{2}}}$

Would you like to explore this more or ask any questions? Here are some related problems:

1. How does the volume of the cylinder change if the radius of the sphere is doubled?
2. What is the relationship between the height of the cylinder and the radius of the sphere?
3. Can we compute the surface area of the smaller cylinder?
4. How does the frustum's volume compare to the original cylinder's volume?
5. What happens if the plane cuts the cylinder at half its height?

Tip: In problems involving inscribed shapes, always think about how the radii and heights of the objects relate geometrically to the enclosing shape (here, the sphere).