The difference between the outer curved surface area and the inner curved surface area of a hollow cylinder is $352$ $\displaystyle cm^{2}.$ If its height is $28$ cm and the volume of material in it is $704$ $\displaystyle cm^{3};$ find its external curved surface area.

The difference between the outer and inner curved surface areas equals 2 * pi * h * (R - r), so 352 = 2 * (22/7) * 28 * (R - r), which gives the radial difference R - r = 2 cm. Using the volume of the material 704 = pi * h * (R^2 - r^2), we factor the difference of squares to get 704 = (22/7) * 28 * 2 * (R + r), meaning the sum of the radii R + r is 4 cm. Solving these two equations yields R = 3 cm and r = 1 cm, making the external curved surface area 2 * (22/7) * 3 * 28, which is 528 square cm.

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