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A cylinder whose height is two thirds of its diameter has the same volume as a sphere of radius 4 cm. The radius of the base of the cylinder will be
- a)4 cm
- b)2 cm
- c)5 cm
- d)8 cm
Correct answer is option 'A'. Can you explain this answer?
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A cylinder whose height is two thirds of its diameter has the same vol...
To solve this problem, we can use the formulas for the volume of a cylinder and the volume of a sphere. Let's break down the solution into steps:
Given information:
- The height of the cylinder is two-thirds of its diameter
- The volume of the cylinder is equal to the volume of a sphere with a radius of 4 cm
Step 1: Calculate the volume of the sphere
The formula for the volume of a sphere is V = (4/3)πr^3, where r is the radius. Since the radius of the sphere is given as 4 cm, we can substitute this value into the formula:
V_sphere = (4/3)π(4^3)
V_sphere = (4/3)π(64)
V_sphere = (4/3)(3.14)(64)
V_sphere ≈ 268.08 cm^3
Step 2: Set up the equation for the volume of the cylinder
The formula for the volume of a cylinder is V = πr^2h, where r is the radius and h is the height. We are given that the height of the cylinder is two-thirds of its diameter. Since the diameter is twice the radius, we can write the height as h = (2/3)(2r) = (4/3)r. Substituting these values into the formula, we get:
V_cylinder = πr^2(4/3)r
V_cylinder = (4/3)πr^3
Step 3: Equate the volumes of the cylinder and the sphere
Since the volume of the cylinder is equal to the volume of the sphere, we can set up the equation:
(4/3)πr^3 = 268.08
Dividing both sides by (4/3)π, we get:
r^3 = 268.08 / (4/3)π
r^3 = 201.06 / π
Taking the cube root of both sides, we find:
r = (201.06/π)^(1/3)
r ≈ 4 cm
Therefore, the radius of the base of the cylinder is approximately 4 cm, which corresponds to option A.
Given information:
- The height of the cylinder is two-thirds of its diameter
- The volume of the cylinder is equal to the volume of a sphere with a radius of 4 cm
Step 1: Calculate the volume of the sphere
The formula for the volume of a sphere is V = (4/3)πr^3, where r is the radius. Since the radius of the sphere is given as 4 cm, we can substitute this value into the formula:
V_sphere = (4/3)π(4^3)
V_sphere = (4/3)π(64)
V_sphere = (4/3)(3.14)(64)
V_sphere ≈ 268.08 cm^3
Step 2: Set up the equation for the volume of the cylinder
The formula for the volume of a cylinder is V = πr^2h, where r is the radius and h is the height. We are given that the height of the cylinder is two-thirds of its diameter. Since the diameter is twice the radius, we can write the height as h = (2/3)(2r) = (4/3)r. Substituting these values into the formula, we get:
V_cylinder = πr^2(4/3)r
V_cylinder = (4/3)πr^3
Step 3: Equate the volumes of the cylinder and the sphere
Since the volume of the cylinder is equal to the volume of the sphere, we can set up the equation:
(4/3)πr^3 = 268.08
Dividing both sides by (4/3)π, we get:
r^3 = 268.08 / (4/3)π
r^3 = 201.06 / π
Taking the cube root of both sides, we find:
r = (201.06/π)^(1/3)
r ≈ 4 cm
Therefore, the radius of the base of the cylinder is approximately 4 cm, which corresponds to option A.
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A cylinder whose height is two thirds of its diameter has the same volume as a sphere of radius 4 cm. The radius of the base of the cylinder will bea)4 cmb)2 cmc)5 cmd)8 cmCorrect answer is option 'A'. Can you explain this answer?
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