7. A solid consisting of a right circular cone of height 120 cm and radius 60 cm standing on
a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of water
such that it touches the bottom. Find the volume of water left in the cylinder, if the radius
of the cylinder is 60 cm and its height is 180 cm.
Answers
Answer :
- The volume of water left in the cylinder is 1131428.6 cm³
Explanation :
Given :
- Radius of the cone, r = 60 cm
- Height of the cone h = 120 cm
- Radius of the hemisphere, r' = 60 cm
- Radius of the cylinder, R = 60 cm
- Height of the cylinder, H = 180 cm.
To find :
- Volume of the water left in the cylinder, V = ?
Knowledge required :
- Formula for volume of a hemisphere :
⠀⠀⠀⠀⠀⠀⠀V = ⅔πr³
[Where :- V = volume of the hemisphere, r = Radius of the hemisphere]
- Formula for volume of a cylinder :
⠀⠀⠀⠀⠀⠀⠀V = πr²h
[Where :- V = volume of the cylinder, r = Radius of the cylinder, h = Height of the cylinder]
- Formula for volume of a cone :
⠀⠀⠀⠀⠀⠀⠀V = ⅓πr²h
[Where :- V = volume of the cone, r = Radius of the cone, h = Height of the cone]
Solution :
Let's find the individual volume of the cylinder, volume of the cone and the volume of the hemisphere.
- Volume of the cylinder :
By using the formula for volume of a cylinder and substituting the values in it, we get :
⠀⠀⠀=> V = πr²h
⠀⠀⠀=> V = 22/7 × 60² × 180
⠀⠀⠀=> V = 14256000/7
⠀⠀⠀⠀⠀⠀⠀⠀⠀∴ V = 14256000/7 cm³
Hence the volume of the cylinder is 14256000/7 cm³
- Volume of the cone :
By using the formula for volume of a cone and substituting the values in it, we get :
⠀⠀⠀=> V = ⅓πr²h
⠀⠀⠀=> V = ⅓ × 22/7 × 60² × 120
⠀⠀⠀=> V = ⅓ × 22/7 × 3600 × 120
⠀⠀⠀=> V = 22/7 × 1200 × 120
⠀⠀⠀=> V = 22/7 × 1200 × 120
⠀⠀⠀=> V = 3168000/7
⠀⠀⠀⠀⠀⠀⠀⠀⠀∴ V = 3168000/7 cm³
Hence the volume of the cone is 3168000/7 cm³
- Volume of the hemisphere :
By using the formula for volume of a cone and substituting the values in it, we get :
⠀⠀⠀=> V = ⅔πr³
⠀⠀⠀=> V = ⅔ × 22/7 × 60³
⠀⠀⠀=> V = ⅔ × 22/7 × 216000
⠀⠀⠀=> V = 2 × 22/7 × 72000
⠀⠀⠀=> V = 3168000/7
⠀⠀⠀⠀⠀⠀⠀⠀⠀∴ V = 3168000/7 cm³
Hence the volume of the hemisphere is 3168000/7 cm³
Volume of the water left in the cylinder :
Volume of the water left in the cylinder = Volume of cylinder - (Volume of cone + volume of hemisphere)
⠀⠀⠀=> V(w) = πr²h - (⅓πr²h + ⅔πr³)
By substituting the values in the above equation, we get :
⠀⠀⠀=> V(w) = 14256000/7 - (3168000/7 + 3168000/7)
⠀⠀⠀=> V(w) = 14256000/7 - (3168000 + 3168000)/7
⠀⠀⠀=> V(w) = 14256000/7 - 6336000/7
⠀⠀⠀=> V(w) = (14256000 - 6336000)/7
⠀⠀⠀=> V(w) = 7920000/7
⠀⠀⠀=> V(w) = 1131428.6 (approx.)
⠀⠀⠀⠀⠀⠀⠀⠀⠀∴ V = 1131428.7 cm³
Therefore,
- Volume of the water left in the cylinder, V = 1131428.6 cm³
Step-by-step explanation:
Given :
- A solid consisting of a right circular cone of height 120 cm and radius 60 cm .
- a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of water such that it touches the bottom.
To Find :
- Find the volume of water left in the cylinder, if the radius of the cylinder is 60 cm and its height is 180 cm.
Solution :
V = πr²H - (1/3πr²h + ⅔ × πr³)
Substitute all values :
V = 22/7 × 60² (180 - ⅓ × 120 - ⅔ × 60)
V = 22/7 × 3600 × (180 - 80)
V = 7,920,000/7
V = 1,131,428.57 cm³
V = 1,131,428.57/ 1000000 m³
V = 1.131 m³
- Hence, the volume of the water left in the cylinder is 1.131 m³.