The sum of the inner and the outer curved surfaces of a hollow metallic cylinder is 1056 cm² and the volume of the material is 1056 cm³. Find its internal and external radii. Given that the height of the cylinder is 21 cm.

Given,

Sum of the inner and the outer curved surfaces of a hollow metallic cylinder = 1056 cm²

∴ 2πrh + 2πRh = 1056

⇒ 2πh(r + R) = 1056

⇒ $\dfrac{44}{7}h(r + R)$ = 1056

⇒ h(r + R) = $\dfrac{1056 \times 7}{44}$

⇒ h(r + R) = 168 ………..(1)

Volume of the material = 1056 cm³

∴ π(R² - r²)h = 1056

⇒ $\dfrac{22}{7} \times (R^2 - r^2)h$ = 1056

⇒ (R² - r²)h = $\dfrac{1056 \times 7}{22}$

⇒ (R² - r²)h = 336 ………(2)

Dividing (2) by (1) we get,

$\Rightarrow \dfrac{(R^2 - r^2)h}{(R + r)h} = \dfrac{336}{168} \\[1em] \Rightarrow \dfrac{(R - r)(R + r)h}{(R + r)h} = 2 \\[1em] \Rightarrow R - r = 2 \\[1em] \Rightarrow R = r + 2$

Substituting value of height and R in equation (1) we get :

⇒ 21(r + r + 2) = 168

⇒ 21(2r + 2) = 168

⇒ 2r + 2 = $\dfrac{168}{21}$

⇒ 2r + 2 = 8

⇒ 2r = 8 - 2

⇒ 2r = 6

⇒ r = $\dfrac{6}{2}$

⇒ r = 3 cm.

⇒ R = r + 2 = 3 + 2 = 5 cm.

Hence, internal radius = 3 cm and external radius = 5 cm.