The difference between the outer curved surface area and the inner curved surface area of a hollow cylinder is 352 cm². If its height is 28 cm and the volume of material in it is 704 cm³; find its external curved surface area.

Let internal radius be r cm and external radius be R cm.

Given,

Difference between the outer curved surface area and the inner curved surface area of a hollow cylinder = 352 cm²

∴ 2πRh - 2πrh = 352

⇒ 2πh(R - r) = 352

⇒ πh(R - r) = 176 ………..(1)

Volume of the material = 704 cm³ (Given)

∴ π(R² - r²)h = 704 ………(2)

Dividing (2) by (1) we get,

$\Rightarrow \dfrac{π(R^2 - r^2)h}{π(R - r)h} = \dfrac{704}{176} \\[1em] \Rightarrow \dfrac{(R - r)(R + r)}{(R - r)} = 4 \\[1em] \Rightarrow R + r = 4 \\[1em] \Rightarrow R = 4 - r$

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

$\Rightarrow \dfrac{22}{7} \times 28 \times (4 - r - r) = 176 \\[1em] \Rightarrow 22 \times 4 \times (4 - 2r) = 176 \\[1em] \Rightarrow 4 - 2r = \dfrac{176}{88} \\[1em] \Rightarrow 4 - 2r = 2 \\[1em] \Rightarrow 2r = 4 - 2 \\[1em] \Rightarrow 2r = 2 \\[1em] \Rightarrow r = 1 \text{ cm}.$

R = 4 - r = 4 - 1 = 3 cm.

External curved surface area = 2πRh

= $2 \times \dfrac{22}{7} \times 3 \times 28$

= 2 × 22 × 3 × 4

= 528 cm².

Hence, external curved surface area = 528 cm².