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A cylinder has the same height as the radius of its base. A hollow sphere has the same outer radius as that of the base of the cylinder while the inner radius is half of the outer radius. Find the ratio of the volumes of the cylinder to the hollow sphere.
- a)5 : 6
- b)7 : 8
- c)6 : 7
- d)3 : 4
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
A cylinder has the same height as the radius of its base. A hollow sph...
Given, height of the cylinder (H) = Base radius (R) of the cylinder
Outer radius of the sphere = Base radius of (R) of the cylinder
Inner radius (r) of the sphere = (outer radius of the sphere)/2
As we know, volume of the cylinder = πR2H
Given, R = H
Volume of the cylinder = π R2 × R = π R3
Volume of the hollow sphere = (4/3) × π (R3 – r3)
Given, r = R/2
Volume of the hollow sphere = (4/3) × π [R3 – (R/2)3] = (4/3) × π × (R3 – R3/8) = (4/3) × π × (7R3/8)
Required ratio of the volumes of the cylinder to the hollow sphere = π R3 : (4/3) × π × (7R3/8)
⇒ 6 : 7
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A cylinder has the same height as the radius of its base. A hollow sph...
To solve this problem, let's first find the volume of the cylinder and the volume of the hollow sphere separately, and then calculate their ratio.
Given information:
- The height of the cylinder is equal to the radius of its base.
- The outer radius of the hollow sphere is equal to the radius of the base of the cylinder.
- The inner radius of the hollow sphere is half the outer radius.
Let's assume:
- The height of the cylinder = h
- The radius of the base of the cylinder = r
- The outer radius of the hollow sphere = R
- The inner radius of the hollow sphere = r'
Volume of the cylinder:
The volume of a cylinder is given by the formula V_cylinder = πr^2h.
Since the height of the cylinder is equal to the radius of its base, h = r.
Therefore, the volume of the cylinder becomes V_cylinder = πr^2r = πr^3.
Volume of the hollow sphere:
The volume of a hollow sphere is given by the formula V_hollow sphere = (4/3)π(R^3 - (r')^3).
Since the inner radius of the hollow sphere is half the outer radius, r' = R/2.
Therefore, the volume of the hollow sphere becomes V_hollow sphere = (4/3)π(R^3 - (R/2)^3).
Calculating the ratio:
To find the ratio of the volumes, we divide the volume of the cylinder by the volume of the hollow sphere:
Ratio = V_cylinder / V_hollow sphere
= πr^3 / [(4/3)π(R^3 - (R/2)^3)]
= 3r^3 / [4(R^3 - R^3/8)]
= 3r^3 / [4(7R^3/8)]
= 3r^3 / (7R^3/2)
= (6r^3 / 7R^3)
Therefore, the ratio of the volumes of the cylinder to the hollow sphere is 6:7, which corresponds to option C.
Given information:
- The height of the cylinder is equal to the radius of its base.
- The outer radius of the hollow sphere is equal to the radius of the base of the cylinder.
- The inner radius of the hollow sphere is half the outer radius.
Let's assume:
- The height of the cylinder = h
- The radius of the base of the cylinder = r
- The outer radius of the hollow sphere = R
- The inner radius of the hollow sphere = r'
Volume of the cylinder:
The volume of a cylinder is given by the formula V_cylinder = πr^2h.
Since the height of the cylinder is equal to the radius of its base, h = r.
Therefore, the volume of the cylinder becomes V_cylinder = πr^2r = πr^3.
Volume of the hollow sphere:
The volume of a hollow sphere is given by the formula V_hollow sphere = (4/3)π(R^3 - (r')^3).
Since the inner radius of the hollow sphere is half the outer radius, r' = R/2.
Therefore, the volume of the hollow sphere becomes V_hollow sphere = (4/3)π(R^3 - (R/2)^3).
Calculating the ratio:
To find the ratio of the volumes, we divide the volume of the cylinder by the volume of the hollow sphere:
Ratio = V_cylinder / V_hollow sphere
= πr^3 / [(4/3)π(R^3 - (R/2)^3)]
= 3r^3 / [4(R^3 - R^3/8)]
= 3r^3 / [4(7R^3/8)]
= 3r^3 / (7R^3/2)
= (6r^3 / 7R^3)
Therefore, the ratio of the volumes of the cylinder to the hollow sphere is 6:7, which corresponds to option C.
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