Question

The difference between outside and inside surfaces of a cylindrical metallic pipe 14cm long is 44 cm². If the pipe is made of 99cm³ of metal,find the outer and inner radii of the pipe​

Answer 1

★ Given:-

• Height of cylindrical pipe = 14 cm long

• Volume of pipe = 99cm³

• Outside surface area - inside surface area = 44cm²

★ To Find:-

• Outer radius of circle

• Inner radius of circle

★ Formula Used:-

• Surface area of Cylinder = 2πrh

• Volume of cylinder = πr²h

• Volume of hollow cylinder = πR²h - πr²h

Here,

▪︎π = 22/7

▪︎r = Radius

▪︎h = Height

★ Solution:-

Let the outer radius be x cm

Let the inner radius be y cm

So,

➥ Outside surface Area

= 2πrh

= 2 × (22/7) × x × 14

= (44 × 14x)÷7

= 88x cm²

➥ Inside surface Area

= 2πrh

= 2 × (22/7) × y × 14

= (44 × 14y)÷7

= 88y cm²

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◈ According to given conditions;

Outside surface - Inside surface = 44

➟ 88x - 88y = 44

➟ 88 ( x - y ) = 44

➟ x - y = 44 ÷ 88

∴ x - y = ½ ------ eq. 1

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➥ External volume

= πr²h

= π × x² × 14

= (22/7) × 14x²

= 44x² cm³

➥ Internal volume

= πr²h

= π × y² × 14

= (22/7) × 14y²

= 44y² cm³

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◈ According to given conditions;

Volume of metal = 99

➟ external volume - inernal volume = 99

➟ 44 x² - 44 y² = 99

➟ 44 ( x² - y² ) = 99

➟ x² - y² = 99 ÷ 44

∴ x² - y² = 9/4 ------- eq. 2

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◈ On Dividing eq. 1 and eq. 2

(x² - y²) ÷ ( x - y ) = (9/4) ÷ (1/2)

➟ ( x - y ) = (9/4) × (2/1)

➟ ( x - y ) = 9/2

∴ x - y = 9/2 ------- eq. 3

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◈ On subtracting eq. 1 from eq. 3

( x + y ) - ( x - y ) = ( 9/2) - ( 1/2)

➟ x + y - x + y = (9 - 1) ÷ 2

➟ x - x + y + y = 8 ÷ 2

➟ 2y = 4

y = 4 ÷ 2

∴ y = inner radius = 2 cm

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◈ Putting value of y in eq. 1

x - y = ½

➟ x - 2 = ½

➟ x = ½ + 2

x = 5/2

∴ x = outer radius = 2.5 cm

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★ Answer:-

• Inner radius = 2 cm

• Outer radius = 2.5 cm

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Answer 2

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Let, R be the external radius and r be the inner radius of the metallic pipe. h=14cm

Outer surface area−inner surface area=44cm2

∴2πRh−2πrh=44cm2

∴R−r=2×722×1444=2×22×1444×7

∴R−r=21.

The volume of metal used =99cm3

∴ External volume − internal volume =99cm3

∴πR2h−πr2h=99cm3

∴πh(R2−r2)=99cm3

∴722×14(R+r)(R−r)=99cm3 [∵a2−b2=(a+b)(a−b)] and [∵R−r=1/2]

22×2(R+r)