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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?
Most Upvoted Answer
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? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about 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? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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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