A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 4 cm and diameter of the base is 8 cm. Determine the volume of the toy. If the cube circumscribes the toy, then find the difference of the volumes of the cube and the toy. Also, find the total surface area of the toy.
Answers
Step-by-step explanation:
For cone,
h = 4cm
r = 4cm
l = √h² + r² = √32 = 4√2 cm
Volume = 1/3πr²h
= 1/3 π x 4 x 4 x 4
For hemisphere,
r = 4cm
Volume = 2/3πr³
= 2/3 π x 4 x 4 x 4
Volume of the toy = 1/3π x 4 x 4 x 4 + 2/3π x 4 x 4 x 4
= 64 x 22/7
= 1408/7 cm³
Volume of cube = a³ = 8³ = 512 cm³
Difference in volumes of the cube and the toy = 512 - 1408/7
= 2176/ 7
= 310.86 cm³
Total surface area of the toy = curved surface area of cone + curved surface area of hemisphere
= πrl + 2πr²
= πr (l + 2r)
=22/7 x 4 (4√2 + 2 x 4)
= 88/7 x 4√2 (1 + √2)
= 171.68 cm² (approx.)
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Given:-
- A solid toy is in the form of a hemisphere surmounted by a right circular cone.
- The height of the cone is 4 cm and diameter of the base is 8 cm.
To find:-
- Determine the volume of the toy.
- The difference of the volumes of the cube and the toy.
- The total surface area of the toy.
Solution:-
According to the question,
Firstly,
- Finding the volume of a toy.
→ Volume of toy = Volume of cone + Volume of hemisphere
→ V = 1/3 πr²h + 2/3 πr³
→ V = πr²[1/3 h + 2/3 r]
→ V = 22/7 × 4 × 4[1/3 × 4 + 2/3 × 4]
→ V = 22/7 × 16[4/3 + 8/3]
→ V = 22/7 × 16[12/31]
→ V = 22/7 × 16 × 4
→ V = 22/7 × 64
→ V = 138.16 cm³
Hence,
- the volume of a toy is 138.16 cm³.
Then,
- Difference of the volumes of the cube and the toy.
→ Dimensions of cube = 8 cm
→ Volume of cube = (8)³ = 512 cm³
→ Volume of cube - Volume of toy
→ 512 - 138.16
→ 373.84 cm³
Now,
→ Slant height of cone => l = √h² + r²
→ l = √(4)² + (4)²
→ l = √16 + 16
→ l = √32
→ l = 4√2 cm
Then,
- Finding total surface area of the toy.
→ TSA = CSA of cone + CSA of hemisphere
→ TSA = πrl + 2πr²
→ TSA = πr(l + 2r)
→ TSA = 22/7 × 4[4√2 + 2(4)]
→ TSA = 22/7 × 4 × 4(√2 + 2)
→ TSA = 352/7 (√2 + 2)
→ TSA = 5.2(√2 + 2) cm²
Hence,
- the total surface of the toy is 5.2(√2 + 2) cm².
