Question

A toy is in the form of a hemisphere surmounted by a right circular cone of the same base radius as that of hemisphere. If the radius of base of the cone is 21cm and the volume is $\frac{2}{3}$ of the volume of the hemisphere, calculate the height of the cone and the surface area of the toy (Use $(\pi=\frac{22}{7})$).

Answer 1

Answer:

The height of the cone is 28 cm and the surface area of a toy is 5082 cm² .

Step-by-step explanation:

SOLUTION :  

Given :  

Radius of cone = Radius of hemisphere, r = 21 cm  

Volume of cone = 2/3 Volume of hemisphere  

⅓ πr²h = ⅔ × ⅔ ×πr³

⅓ r²h = 4/9 ×r³

⅓ × 21² × h = 4/9 × 21³

h = (3 × 4 × 21³)/(9 × 21²)

h = (4 × 21)/3 = 4 × 7 = 28 cm

h = 28 cm

Hence, the height of the cone is 28 cm .

Slant height , l = √(h² + r²)

l = √(28² + 21²)  

l = √(784 + 441)  

l = √(1225)

l = 35 cm

Curved surface area of a toy ,S = Curved surface area of hemisphere + curved surface area of cone

S = 2πr² + πrl

S = πr(2r + l)

S = 22/7 × 21 (2 × 21 + 35)

S = 66 × (42 + 35)

S = 66 × 77

S = 5082 cm²

Hence, the height of the cone is 28 cm and the surface area of a toy is 5082 cm²

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Answer 2

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The height of the cone is 28 cm

and

surface area of the toy is 5082 cm²

Step-by-step explanation:

Given,

Radius of cone = Radius of hemisphere, r = 21 cm  

Volume of cone = 2/3 Volume of hemisphere  

=> ⅓ πr²h = ⅔ × ⅔ ×πr³

=> ⅓ r²h = 4/9 ×r³

putting the value of 'r'

we get,

=> ⅓ × 21² × h = 4/9 × 21³

=> h = (3 × 4 × 21³)/(9 × 21²)

=> h = (4 × 21)/3 = 4 × 7 = 28 cm

=> h = 28 cm

Hence,

the height of the cone is 28 cm .

therefore,

Slant height , l = √(h² + r²)

=> l = √(28² + 21²)  

=> l = √(784 + 441)  

=> l = √(1225)

=> l = 35 cm

also,

Curved surface area of a toy ,S = Curved surface area of hemisphere + curved surface area of cone

=> S = 2πr² + πrl

=> S = πr(2r + l)

=> S = 22/7 × 21 (2 × 21 + 35)

=> S = 66 × (42 + 35)

=> S = 66 × 77

=> S = 5082 cm²

Hence,

the height of the cone is 28 cm

and

the surface area of a toy is 5082 cm²