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 ?

The volume of the solid cylinder must equal the volume of the hollow cylinder. The radius of the solid cylinder is $1$ cm. The external radius of the hollow cylinder is $6$ cm and its internal radius is $6 - 0.25 = 5.75$ cm. Setting the volumes equal gives $\pi \times 1^2 \times L = \pi \times (6^2 - 5.75^2) \times 15$. Solving for $L$ results in $L = (36 - 33.0625) \times 15 = 2.9375 \times 15 = 44.0625$ cm. The required length is 44.0625 cm.

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