The difference between the outer and the inner curved surface areas of a hollow cylinder, 14 cm long, is 88 sq. cm. Find the outer and the inner radii of the cylinder, given that the volume of metal used is 176 cu. cm.

Given,

Height of cylinder (H) = 14 cm

Let outer radius of cylinder be R cm and inner radius be r cm.

Given,

Difference between the outer and the inner curved surface areas of a hollow cylinder is 88 sq. cm.

$\therefore 2πRH - 2πrH = 88 \\[1em] \Rightarrow 2πH(R - r) = 88 \\[1em] \Rightarrow 2 \times \dfrac{22}{7} \times 14 (R - r) = 88 \\[1em] \Rightarrow 88(R - r) = 88 \\[1em] \Rightarrow R - r = 1 \space …………(1)$

Given,

Volume of metal = 176 cu. cm

$\therefore π(R^2 - r^2)H = 176 \\[1em] \Rightarrow \dfrac{22}{7} \times (R^2 - r^2) \times 14 = 176 \\[1em] \Rightarrow 44(R^2 - r^2) = 176 \\[1em] \Rightarrow (R^2 - r^2) = \dfrac{176}{44} \\[1em] \Rightarrow (R^2 - r^2) = 4 \\[1em] \Rightarrow (R - r)(R + r) = 4 \\[1em] \Rightarrow 1 \times (R + r) = 4 \text{ …….[From (1)]} \\[1em] \Rightarrow (R + r) = 4 ………..(2)$

Adding equation (1) and (2), we get :

⇒ R - r + R + r = 1 + 4

⇒ 2R = 5

⇒ R = $\dfrac{5}{2}$ = 2.5 cm

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

⇒ 2.5 - r = 1

⇒ r = 2.5 - 1 = 1.5 cm

Hence, outer radii = 2.5 cm and inner radii = 1.5 cm.