What length of solid cylinder $2$ cm in diameter must be taken to cast into a hollow cylinder of external diameter $12$ cm, $0.25$ cm thick and $15$ cm long?

### Concept & Conservation of Volume When a solid object is melted and recast into another shape, the total volume of the material remains unchanged. We equate the volume of the solid cylinder to the volume of the hollow cylinder. $$ V_{\text{solid}} = V_{\text{hollow}} $$ $$ \pi \times r_1^2 \times h_1 = \pi \times (R_2^2 - r_2^2) \times h_2 $$ ### Step-by-Step Solution - **Given for Solid Cylinder:** - Diameter $= 2$ cm $\implies$ radius $r_1 = 1$ cm. - Let its length be $h_1 = L$. - **Given for Hollow Cylinder:** - External diameter $= 12$ cm $\implies$ outer radius $R_2 = 6$ cm. - Thickness $= 0.25$ cm $\implies$ inner radius $r_2 = 6 - 0.25 = 5.75$ cm. - Length $h_2 = 15$ cm. - **Calculation:** - Equate the volumes and cancel out $\pi$ from both sides: - $1^2 \times L = (6^2 - 5.75^2) \times 15$ - Using difference of squares for $(6^2 - 5.75^2) = (6 - 5.75)(6 + 5.75) = 0.25 \times 11.75 = 2.9375$. - $L = 2.9375 \times 15 = 44.0625$ cm. ### Exam Strategy & Shortcut Always leave $\pi$ as a symbol in melting/recasting problems because it will almost always cancel out on both sides of the equation. ### Common Pitfall Miscalculating the inner radius by subtracting the thickness from the diameter instead of the radius. ### Final Answer Therefore, the correct answer is **44.0625 cm**.