Volume Calculation for Hollow Sphere

The volume of a hollow spherical shell is given by the formula:

$ V_{\text{shell}} = \frac{4}{3}\pi (R^3 - r^3) $

Where $R$ is the external radius and $r$ is the internal radius. From the question, we have the radii as 6 cm and 4 cm. Since the external radius must be greater than the internal radius, we assume $ R = 6 \text{ cm} $ and $ r = 4 \text{ cm} $.

Substituting the values:

$ V_{\text{shell}} = \frac{4}{3}\pi (6^3 - 4^3) $

$ V_{\text{shell}} = \frac{4}{3}\pi (216 - 64) $

$ V_{\text{shell}} = \frac{4}{3}\pi (152) \text{ cm}^3 $

Volume Calculation for Solid Cylinder

The volume of a solid cylinder is given by the formula:

$ V_{\text{cylinder}} = \pi r_{\text{cyl}}^2 h $

Where $r_cyl$ is the radius and $h$ is the height. The diameter $d$ is $ 2 r_{\text{cyl}} $, so $ r_{\text{cyl}} = \frac{d}{2} $. The height is given as $ h = \frac{8}{3} \text{ cm} $.

Substituting these into the volume formula:

$ V_{\text{cylinder}} = \pi \left(\frac{d}{2}\right)^2 \left(\frac{8}{3}\right) $

$ V_{\text{cylinder}} = \pi \left(\frac{d^2}{4}\right) \left(\frac{8}{3}\right) $

$ V_{\text{cylinder}} = \frac{2 \pi d^2}{3} \text{ cm}^3 $

Equating Volumes to Find Cylinder Diameter

The material from the hollow sphere is melted and recast into a solid cylinder, so their volumes must be equal:

$ V_{\text{shell}} = V_{\text{cylinder}} $

$ \frac{4}{3}\pi (152) = \frac{2 \pi d^2}{3} $

Cancel out $ \frac{2\pi}{3} $ from both sides:

$ 2 \times 152 = d^2 $

$ 304 = d^2 $

Solve for the diameter $d$:

$ d = \sqrt{304} $

Simplify the square root:

$ d = \sqrt{16 \times 19} $

$ d = 4\sqrt{19} \text{ cm} $