Question

The difference between inside and outside surfaces of a cylindrical tube 14 cm long, is 88 cm². If the volume of the tube is 176 cm³, find the inner and outer radii of the tube.

Answer 1

Let the inner and outer radii of the tube be r cm and R cm, respectively.

Length of the cylindrical tube, h = 14 cm

Outer curved surface of the cylinder − Inner curved surface of the cylinder = 88 cm² (Given)

$$\begin{gathered} \therefore 2\mathrm{\pi }Rh-2\mathrm{\pi }rh=88 \\ \Rightarrow 2\times \frac{22}{7}\times 14\times \left(R\mathit{-}r\right)=88 \\ \Rightarrow R\mathit{-}r=\frac{88\times 7}{2\times 22\times 14} \\ \Rightarrow R-r=1\mathrm{cm}.....\left(1\right) \end{gathered}$$
Volume of the tube = 176 cm³ (Given)

$$\begin{gathered} \therefore \mathrm{\pi }\left(R^2\mathit{-}{r}^{\mathit{2}}\right)h=176 \\ \Rightarrow \frac{22}{7}\times \left(R\mathit{-}r\right)\times \left(R+r\right)\times 14=176 \\ \Rightarrow R+r=\frac{176\times 7}{22\times 1\times 14}\left[\mathrm{Using}\left(1\right)\right] \\ \Rightarrow R+r=4\mathrm{cm}.....\left(2\right) \end{gathered}$$
Adding (1) and (2), we get

2R = 5

⇒ R = 2.5 cm

Putting R = 2.5 cm in (2), we get

2.5 + r = 4

⇒ r = 4 − 2.5 = 1.5 cm

Thus, the inner and outer radii of the tube are 1.5 cm and 2.5 cm, respectively.