Question

The difference between outside and inside surface areas of cylindrical metallic pipe 14 cm long is 44 m². If the pipe is made of 99 cm³ of metal, find the outer and inner radii of the pipe.

Answer 1

Given: 

The difference between outside and inside surface areas of cylindrical metallic pipe 14 cm long is 44 m² and the pipe is made of 99 cm³ of metal. 

To find: 

the outer and inner radii of the pipe. 

Solution:

Let the inner radius of pipe = r₁ 

And the radius of outer cylinder = r₂ 

Length of cylinder (h) = 14 cm 

Surface area of hollow cylinder =2πh (r₂ – r₁) 

Given surface area of cylinder = 44 m². 

⇒ 2π h(r₂ –r₁) = 44 

⇒ 2π (14) (r₂ –r₁) = 44 

⇒ (r₂ – r₁) = \(\frac{44}{28π}\)

⇒ (r₂ –r₁) = 1/2 ……. (1) 

Given volume of a hollow cylinder = 99 cm³ 

Volume of a hollow cylinder = π h(r₂² - r₁² ) 

⇒ π h (r₂² - r₁²) = 99 

⇒ 14π (r₂² - r₁²⁾ = 99 

Apply the formula a² - b² = ( a - b ) ( a + b ) in (r₂² - r₁²). 

⇒ 14π (r₂ + r₁) (r₂ – r₁) = 99 

Put the value of r2 - r₁ from (1). 

⇒ 14π (r₂ + r₁) \(\frac{1}2\) = 99

⇒ (r₂ + r₁) = \(\frac{9}2\)……………… (2) 

Equating (1) & (2) equations we get 

r₂ = \(\frac{5}2\) cm 

Substituting r₂ value in (1)

= \(\frac{5}2\) - r₁ = \(\frac{1}2\)

⇒ \(\frac{5}2\) - \(\frac{1}2\) = r₁

⇒ r₁ = \(\frac{5 - 1}2\)

⇒ r₁ = \(\frac{4}2\)

⇒ r₁ = 2 cm 

∴ Inner radius of pipe (r₁) = 2 cm 

Radius of outer cylinder (r2) = \(\frac{5}2\) cm