What length of solid cylinder 2 cm in diameter must be taken to recast into a hollow cylinder of external diameter 20 cm, 0.25 cm thick and 15 cm long?

Given,

External diameter of hollow cylinder = 20 cm

So, external radius (R) = $\dfrac{20}{2}$ = 10 cm.

Thickness = 0.25 cm

Hence, the internal radius (r) = (10 - 0.25) = 9.75 cm.

Length of cylinder (h) = 15 cm

By formula,

$\text{Volume of material} = πh(R^2 - r^2) \\[1em] = π \times 15 \times [10^2 - (9.75)^2] \\[1em] = π \times 15 \times [100 - 95.0625] \\[1em] = π \times 15 \times 4.9375.$

Given,

Diameter of the solid cylinder = 2 cm

so, radius (a) = $\dfrac{2}{2}$ = 1 cm

Let h be the length of the solid cylinder,

Volume = πa²h = π(1)²h = πh cm³.

In order for recasting, volume of solid cylinder must be equal to volume of material of hollow cylinder .

∴ πh = 15 π x 4.9375

⇒ h = 15 x 4.9375

⇒ h = 74.0625 cm.

Hence, the length of solid cylinder = 74.06 cm.