Question

Q 21. The internal and external diameter of a hollow hemispherical vessel are
21 cm and 28 cm respectively Find:
(i) Internal curved surface area
(ii) External curved surface area
(iii) Total surface area
(iv) Volume of material of the vessel​

Answer 1

$\underline{\underline{\sf{\maltese\:\:Question}}}$

⭐ The Internal and External diameter of a hollow hemispherical vessel are

21 cm and 28 cm respectively. Find :

• Internal curved surface area

• External curved surface area

• Total Surface Area

• Volume of material of the vessel​

$\underline{\underline{\sf{\maltese\:\:Given}}}$

• The Internal Diameter of Hollow Hemispherical Vessel = 21 cm

• The External Diameter of Hollow Hemispherical Vessel = 28 cm

$\underline{\underline{\sf{\maltese\:\:To\:Find}}}$

• Internal curved surface area

• External curved surface area

• Total Surface Area

• Volume of material of the vessel​

$\underline{\underline{\sf{\maltese\:\:Answer}}}$

• Internal Curved Surface Area = 693 cm²

• External Curved Surface Area = 1232 cm²

• Total Surface Area = 2194.5 cm²

• Volume of material of the vessel​ = 3323.83 cm³

$\underline{\underline{\sf{\maltese\:\:Calculations}}}$

We need to know about some basic terms before going to Calculations

• Diameter : The center of a circle is the midpoint of its diameter, It divides the circle into two equal parts
• Radius : The distance from the center to the circumference of a circle

Also :

• Diameter = 2 × Radius
• Radius = Diameter/2

_______________________________________________________

Internal Curved Surface Area = 2πr²

⇒ Internal Curved Surface Area = 2 × π × r²

Since Radius = diameter/2

⇒ Internal Curved Surface Area = 2 × π × (diameter/2)²

⇒ Internal Curved Surface Area = 2 × π × (21 cm/2)²

⇒ Internal Curved Surface Area = 2 × π × (21²/2²) cm²

⇒ Internal Curved Surface Area = 2 × π × ((21 × 21)/(2 × 2)) cm²

We Know π = 22/7

⇒ Internal Curved Surface Area = 2 × 22/7 × ((21 × 21)/(2 × 2)) cm²

⇒ Internal Curved Surface Area = 2 × 22/1 × ((3 × 21)/(2 × 2)) cm²

⇒ Internal Curved Surface Area = 2 × 22/1 × (63/4) cm²

⇒ Internal Curved Surface Area = (2 × 22)/1 × (63/4) cm²

⇒ Internal Curved Surface Area = 44/1 × (63/4) cm²

⇒ Internal Curved Surface Area = 44 × 63/4 cm²

⇒ Internal Curved Surface Area = 11 × 63/1 cm²

⇒ Internal Curved Surface Area = 693 cm²

_______________________________________________________

External Curved Surface Area = 2πR²

⇒ External Curved Surface Area = 2 × π × R²

Since Radius = Diameter/2

⇒ External Curved Surface Area = 2 × π × (Diameter/2)²

⇒ External Curved Surface Area = 2 × π × (28 cm/2)²

⇒ External Curved Surface Area = 2 × π × (28²/2²) cm²

⇒ External Curved Surface Area = 2 × π × ((28 × 28)/(2 × 2)) cm²

We Know π = 22/7

⇒ External Curved Surface Area = 2 × 22/7 × ((28 × 28)/(2 × 2)) cm²

⇒ External Curved Surface Area = 2 × 22/1 × ((4 × 28)/(2 × 2)) cm²

⇒ External Curved Surface Area = 2 × 22/1 × (112/4) cm²

⇒ External Curved Surface Area = 2 × 22 × (112/4) cm²

⇒ External Curved Surface Area = 44 × (112/4) cm²

⇒ External Curved Surface Area = 11 × (112/1) cm²

⇒ External Curved Surface Area = 11 × 112 cm²

⇒ External Curved Surface Area = 1232 cm²

_______________________________________________________

Total Surface Area

= External Curved Surface Area + Internal Curved Surface Area + π(R² - r²)

⇒ Total Surface Area = 693 cm² + 1232 cm² + π(R² - r²)

⇒ Total Surface Area = 1925 cm² + π(R² - r²)

We Know π = 22/7

⇒ Total Surface Area = 1925 cm² + 22/7(R² - r²)

Since Radius = Diameter/2

⇒ Total Surface Area = 1925 cm² + 22/7((Diameter/2)² - (diameter/2)²)

⇒ Total Surface Area = 1925 cm² + 22/7((28 cm/2)² - (21 cm/2)²)

⇒ Total Surface Area = 1925 cm² + 22/7((14 cm)² - (441 /4) cm²)

⇒ Total Surface Area = 1925 cm² + 22/7(196 cm² - 110.25 cm²)

⇒ Total Surface Area = 1925 cm² + 22/7(85.75 cm²)

⇒ Total Surface Area = 1925 cm² + (22 × 85.75)/7 cm²

⇒ Total Surface Area = 1925 cm² + 1886.5/7 cm²

⇒ Total Surface Area = 1925 cm² + 269.5 cm²

⇒ Total Surface Area = 2194.5 cm²

_______________________________________________________

Volume of material of the vessel​ = 2/3π(R³ - r³)

⇒ Volume of material of the vessel​ = 2/3 × π × (R³ - r³)

We Know π = 22/7

⇒ Volume of material of the vessel​ = 2/3 × 22/7 × (R³ - r³)

⇒ Volume of material of the vessel​ = 44/21 × (R³ - r³)

Since Radius = Diameter/2

⇒ Volume of material of the vessel​ = 44/21 × ((Diameter/2)³ - (diameter/2)³)

⇒ Volume of material of the vessel​ = 44/21 × ((28 cm/2)³ - (21 cm/2)³)

⇒ Volume of material of the vessel​ = 44/21 × ((14 cm)³ - (21/2)³ cm³)

⇒ Volume of material of the vessel​ = 44/21 × ((14 cm)³ - (9261/8) cm³)

⇒ Volume of material of the vessel​ = 44/21 × (2744 cm³ - (9261/8) cm³)

⇒ Volume of material of the vessel​ = 44/21 × (2744 cm³ - 1157.625 cm³)

⇒ Volume of material of the vessel​ = 44/21 × (1586.375 cm³)

⇒ Volume of material of the vessel​ = 44/21 × (1586.375) cm³

⇒ Volume of material of the vessel​ = (44 × 1586.375)/21 cm³

⇒ Volume of material of the vessel​ = 69800.5/21 cm³

⇒ Volume of material of the vessel​ = 3323.83 cm³

_______________________________________________________

Request :  

If there is any difficulty viewing this answer in app, Kindly see this answer at Web (https://brainly.in/) for clear steps and understanding .

See the answer at :

https://brainly.in/question/31146091