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A sphere of diameter 6 cm is dropped into a right circular cylindrical vessel, partly filled with water. The diameter of the cylindrical vessel is 12 cm. If the sphere is completely submerged in water, by how much will the level of water rise in the cylindrical vessel?
- a)0.8 cm
- b)1 cm
- c)0.1 cm
- d)2 cm
Correct answer is option 'B'. Can you explain this answer?
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A sphere of diameter 6 cm is dropped into a right circular cylindrical...
Given information:
- Diameter of the sphere = 6 cm
- Diameter of the cylindrical vessel = 12 cm
To find:
- By how much will the level of water rise in the cylindrical vessel when the sphere is completely submerged.
Let's solve this problem step by step:
1. Calculating the volume of the sphere:
- The formula for the volume of a sphere is V = (4/3)πr³, where r is the radius of the sphere.
- Diameter of the sphere = 6 cm, so the radius (r) = 6/2 = 3 cm.
- Substituting the value of r in the formula, we get V = (4/3)π(3)³ = 36π cm³.
2. Calculating the volume of the cylindrical vessel:
- The formula for the volume of a cylinder is V = πr²h, where r is the radius of the cylinder and h is the height.
- Diameter of the cylindrical vessel = 12 cm, so the radius (r) = 12/2 = 6 cm.
- Let the initial height of the water in the cylindrical vessel be h1.
- Substituting the values of r and h1 in the formula, we get V = π(6)²h1 = 36πh1 cm³.
3. Calculating the new volume of the cylindrical vessel after the sphere is submerged:
- When the sphere is completely submerged in water, its volume will be equal to the volume of water displaced.
- So, the new volume of the cylindrical vessel will be the sum of the initial volume of water (36πh1 cm³) and the volume of the sphere (36π cm³), which is equal to 72πh1 cm³.
4. Calculating the new height of the water in the cylindrical vessel:
- The formula for the volume of a cylinder is V = πr²h, where r is the radius of the cylinder and h is the height.
- Let the new height of the water in the cylindrical vessel be h2.
- Substituting the values of r and h2 in the formula, we get 72πh1 = π(6)²h2.
- Simplifying the equation, we get 36h1 = 36h2.
- Dividing both sides of the equation by 36, we get h1 = h2.
Therefore, the level of water will rise by the same height as the radius of the sphere, which is 3 cm. So, the correct answer is option B) 1 cm.
- Diameter of the sphere = 6 cm
- Diameter of the cylindrical vessel = 12 cm
To find:
- By how much will the level of water rise in the cylindrical vessel when the sphere is completely submerged.
Let's solve this problem step by step:
1. Calculating the volume of the sphere:
- The formula for the volume of a sphere is V = (4/3)πr³, where r is the radius of the sphere.
- Diameter of the sphere = 6 cm, so the radius (r) = 6/2 = 3 cm.
- Substituting the value of r in the formula, we get V = (4/3)π(3)³ = 36π cm³.
2. Calculating the volume of the cylindrical vessel:
- The formula for the volume of a cylinder is V = πr²h, where r is the radius of the cylinder and h is the height.
- Diameter of the cylindrical vessel = 12 cm, so the radius (r) = 12/2 = 6 cm.
- Let the initial height of the water in the cylindrical vessel be h1.
- Substituting the values of r and h1 in the formula, we get V = π(6)²h1 = 36πh1 cm³.
3. Calculating the new volume of the cylindrical vessel after the sphere is submerged:
- When the sphere is completely submerged in water, its volume will be equal to the volume of water displaced.
- So, the new volume of the cylindrical vessel will be the sum of the initial volume of water (36πh1 cm³) and the volume of the sphere (36π cm³), which is equal to 72πh1 cm³.
4. Calculating the new height of the water in the cylindrical vessel:
- The formula for the volume of a cylinder is V = πr²h, where r is the radius of the cylinder and h is the height.
- Let the new height of the water in the cylindrical vessel be h2.
- Substituting the values of r and h2 in the formula, we get 72πh1 = π(6)²h2.
- Simplifying the equation, we get 36h1 = 36h2.
- Dividing both sides of the equation by 36, we get h1 = h2.
Therefore, the level of water will rise by the same height as the radius of the sphere, which is 3 cm. So, the correct answer is option B) 1 cm.
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Community Answer
A sphere of diameter 6 cm is dropped into a right circular cylindrical...
Radius of the sphere = 6/2 = 3 cm
Radius of the cylinder = 12/2 = 6cm
Volume of the sphere
New volume of the cylindrical vessel - Initial volume of the cylindrical vessel = Volume of the sphere
Initial height = h cm
Final height = H cm
πr2H - πr2h = 36π
Radius of the cylinder = 12/2 = 6cm
Volume of the sphere
New volume of the cylindrical vessel - Initial volume of the cylindrical vessel = Volume of the sphere
Initial height = h cm
Final height = H cm
πr2H - πr2h = 36π
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