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A cylindrical vessel 60 cm in diameter is partially filled with water. A sphere 30 cm in diameter is gently dropped into the vessel and is completely immersed. To what further height will the water in the cylinder rise?
- a)20 cm
- b)15 cm
- c)10 cm
- d)5 cm
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
A cylindrical vessel 60 cm in diameter is partially filled with water....
Volume of water rise = Volume of sphere
⇒ Volume of water rise = 4/3πR3 [where R is the radius of the sphere]
⇒ Volume of water rise = 4/3πR3 [where R is the radius of the sphere]
⇒ Volume of water rise = 4/3 × π × 153 = 4500π cm3
Height of rise = Volume of water rise/Area of base of cylinder
⇒ Height of rise = 4500π/(πr2)
⇒ Height of rise = 4500/(30)2 = 5 cm
∴ Water will rise further 5 cm in the cylinder
Height of rise = Volume of water rise/Area of base of cylinder
⇒ Height of rise = 4500π/(πr2)
⇒ Height of rise = 4500/(30)2 = 5 cm
∴ Water will rise further 5 cm in the cylinder
Most Upvoted Answer
A cylindrical vessel 60 cm in diameter is partially filled with water....
Given information:
- Diameter of the cylindrical vessel = 60 cm
- Diameter of the sphere = 30 cm
To find: The further height to which the water in the cylinder will rise.
Solution:
1. We can start by finding the volume of the sphere. The formula for the volume of a sphere is given by V = (4/3)πr³, where r is the radius of the sphere. Since the diameter is given, we can find the radius by dividing the diameter by 2.
- Diameter of the sphere = 30 cm
- Radius of the sphere (r) = 30 cm / 2 = 15 cm
2. Substituting the value of the radius into the volume formula, we get:
V = (4/3)π(15 cm)³
V = (4/3)π(15 cm)(15 cm)(15 cm)
V ≈ 14137.17 cm³
3. Now, we need to find the volume of the water that is displaced when the sphere is dropped into the cylindrical vessel. Since the sphere is completely immersed, the volume of the water displaced is equal to the volume of the sphere.
- Volume of water displaced = 14137.17 cm³
4. The volume of a cylinder is given by V = πr²h, where r is the radius of the cylinder and h is the height. We can rearrange this formula to solve for h:
h = V / (πr²)
5. Substituting the values into the formula, we get:
h = 14137.17 cm³ / (π(30 cm)²)
h ≈ 5 cm
Therefore, the water in the cylinder will rise by approximately 5 cm further when the sphere is dropped into it. Hence, option D is the correct answer.
- Diameter of the cylindrical vessel = 60 cm
- Diameter of the sphere = 30 cm
To find: The further height to which the water in the cylinder will rise.
Solution:
1. We can start by finding the volume of the sphere. The formula for the volume of a sphere is given by V = (4/3)πr³, where r is the radius of the sphere. Since the diameter is given, we can find the radius by dividing the diameter by 2.
- Diameter of the sphere = 30 cm
- Radius of the sphere (r) = 30 cm / 2 = 15 cm
2. Substituting the value of the radius into the volume formula, we get:
V = (4/3)π(15 cm)³
V = (4/3)π(15 cm)(15 cm)(15 cm)
V ≈ 14137.17 cm³
3. Now, we need to find the volume of the water that is displaced when the sphere is dropped into the cylindrical vessel. Since the sphere is completely immersed, the volume of the water displaced is equal to the volume of the sphere.
- Volume of water displaced = 14137.17 cm³
4. The volume of a cylinder is given by V = πr²h, where r is the radius of the cylinder and h is the height. We can rearrange this formula to solve for h:
h = V / (πr²)
5. Substituting the values into the formula, we get:
h = 14137.17 cm³ / (π(30 cm)²)
h ≈ 5 cm
Therefore, the water in the cylinder will rise by approximately 5 cm further when the sphere is dropped into it. Hence, option D is the correct answer.
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A cylindrical vessel 60 cm in diameter is partially filled with water. A sphere 30 cm in diameter is gently dropped into the vessel and is completely immersed. To what further height will the water in the cylinder rise?a)20 cmb)15 cmc)10 cmd)5 cmCorrect answer is option 'D'. Can you explain this answer?
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