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DIRECTIONS for questions: Select the correct alternative from the given choices.
Consider two cubes of equal volume. In one of the cubes, the biggest possible sphere is inscribed. The other cube is cut into exactly 64 identical small cubes. In each of them, the biggest possible sphere is placed. The volume of the sphere in the first cube is denoted by V1. The total volume of the spheres in the small cubes is denoted by V2. Find V1 : V2.
- a)4 : 3
- b)1 : 1
- c)3 : 4
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
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DIRECTIONS for questions: Select the correct alternative from the giv...
For any cube, the ratio of the volume of the largest possible sphere that can be placed in it to its volume is constant.
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As the sum of the volumes of all the small cubes is equal to the volume of the first cube, the sum of the volumes of the spheres in these cubes is equal to the volume of the sphere is the first cube.
∴The required ratio is 1 :1.
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DIRECTIONS for questions: Select the correct alternative from the giv...
Given:
- Two cubes of equal volume
- In one cube, the biggest possible sphere is inscribed
- The other cube is cut into 64 identical small cubes
- In each small cube, the biggest possible sphere is placed
- The volume of the sphere in the first cube is V1
- The total volume of the spheres in the small cubes is V2
To find: V1 : V2
Solution:
Let's assume the side length of each cube to be 'a'.
Volume of each cube = side length^3 = a^3
Volume of the bigger cube (with inscribed sphere) = Volume of the smaller cube (cut into small cubes)
a^3 = (a/4)^3 * 64
Simplifying the equation:
a^3 = (a^3/64) * 64
a^3 = a^3
This equation holds true for any value of 'a'.
Now, let's calculate the volume of the sphere in the bigger cube (V1):
The diameter of the sphere is equal to the side length of the bigger cube (a).
Radius (r) of the sphere = diameter/2 = a/2
Volume of the sphere (V1) = (4/3) * π * r^3
V1 = (4/3) * π * (a/2)^3
V1 = (4/3) * π * (a^3/8)
V1 = (π/6) * a^3
Next, let's calculate the total volume of the spheres in the small cubes (V2):
Since there are 64 small cubes, there are 64 spheres.
The diameter of each sphere is equal to the side length of each small cube (a/4).
Radius (r) of each sphere = diameter/2 = (a/4)/2 = a/8
Volume of each sphere = (4/3) * π * r^3
V2 = 64 * [(4/3) * π * (a/8)^3]
V2 = (64/6) * π * (a^3/512)
V2 = (32/3) * π * (a^3/512)
V2 = (π/48) * a^3
Therefore, V1 : V2 = [(π/6) * a^3] : [(π/48) * a^3]
V1 : V2 = 48 : 6
V1 : V2 = 8 : 1
Hence, the correct answer is option 'B' - 1 : 1.
- Two cubes of equal volume
- In one cube, the biggest possible sphere is inscribed
- The other cube is cut into 64 identical small cubes
- In each small cube, the biggest possible sphere is placed
- The volume of the sphere in the first cube is V1
- The total volume of the spheres in the small cubes is V2
To find: V1 : V2
Solution:
Let's assume the side length of each cube to be 'a'.
Volume of each cube = side length^3 = a^3
Volume of the bigger cube (with inscribed sphere) = Volume of the smaller cube (cut into small cubes)
a^3 = (a/4)^3 * 64
Simplifying the equation:
a^3 = (a^3/64) * 64
a^3 = a^3
This equation holds true for any value of 'a'.
Now, let's calculate the volume of the sphere in the bigger cube (V1):
The diameter of the sphere is equal to the side length of the bigger cube (a).
Radius (r) of the sphere = diameter/2 = a/2
Volume of the sphere (V1) = (4/3) * π * r^3
V1 = (4/3) * π * (a/2)^3
V1 = (4/3) * π * (a^3/8)
V1 = (π/6) * a^3
Next, let's calculate the total volume of the spheres in the small cubes (V2):
Since there are 64 small cubes, there are 64 spheres.
The diameter of each sphere is equal to the side length of each small cube (a/4).
Radius (r) of each sphere = diameter/2 = (a/4)/2 = a/8
Volume of each sphere = (4/3) * π * r^3
V2 = 64 * [(4/3) * π * (a/8)^3]
V2 = (64/6) * π * (a^3/512)
V2 = (32/3) * π * (a^3/512)
V2 = (π/48) * a^3
Therefore, V1 : V2 = [(π/6) * a^3] : [(π/48) * a^3]
V1 : V2 = 48 : 6
V1 : V2 = 8 : 1
Hence, the correct answer is option 'B' - 1 : 1.
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DIRECTIONS for questions: Select the correct alternative from the given choices.Consider two cubes of equal volume. In one of the cubes, the biggest possible sphere is inscribed. The other cube is cut into exactly 64 identical small cubes. In each of them, the biggest possible sphere is placed. The volume of the sphere in the first cube is denoted by V1. The total volume of the spheres in the small cubes is denoted by V2. Find V1 : V2.a) 4 : 3b) 1 : 1c) 3 : 4d) None of theseCorrect answer is option 'B'. Can you explain this answer?
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DIRECTIONS for questions: Select the correct alternative from the given choices.Consider two cubes of equal volume. In one of the cubes, the biggest possible sphere is inscribed. The other cube is cut into exactly 64 identical small cubes. In each of them, the biggest possible sphere is placed. The volume of the sphere in the first cube is denoted by V1. The total volume of the spheres in the small cubes is denoted by V2. Find V1 : V2.a) 4 : 3b) 1 : 1c) 3 : 4d) None of theseCorrect answer is option 'B'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about DIRECTIONS for questions: Select the correct alternative from the given choices.Consider two cubes of equal volume. In one of the cubes, the biggest possible sphere is inscribed. The other cube is cut into exactly 64 identical small cubes. In each of them, the biggest possible sphere is placed. The volume of the sphere in the first cube is denoted by V1. The total volume of the spheres in the small cubes is denoted by V2. Find V1 : V2.a) 4 : 3b) 1 : 1c) 3 : 4d) None of theseCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for DIRECTIONS for questions: Select the correct alternative from the given choices.Consider two cubes of equal volume. In one of the cubes, the biggest possible sphere is inscribed. The other cube is cut into exactly 64 identical small cubes. In each of them, the biggest possible sphere is placed. The volume of the sphere in the first cube is denoted by V1. The total volume of the spheres in the small cubes is denoted by V2. Find V1 : V2.a) 4 : 3b) 1 : 1c) 3 : 4d) None of theseCorrect answer is option 'B'. Can you explain this answer?.
DIRECTIONS for questions: Select the correct alternative from the given choices.Consider two cubes of equal volume. In one of the cubes, the biggest possible sphere is inscribed. The other cube is cut into exactly 64 identical small cubes. In each of them, the biggest possible sphere is placed. The volume of the sphere in the first cube is denoted by V1. The total volume of the spheres in the small cubes is denoted by V2. Find V1 : V2.a) 4 : 3b) 1 : 1c) 3 : 4d) None of theseCorrect answer is option 'B'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about DIRECTIONS for questions: Select the correct alternative from the given choices.Consider two cubes of equal volume. In one of the cubes, the biggest possible sphere is inscribed. The other cube is cut into exactly 64 identical small cubes. In each of them, the biggest possible sphere is placed. The volume of the sphere in the first cube is denoted by V1. The total volume of the spheres in the small cubes is denoted by V2. Find V1 : V2.a) 4 : 3b) 1 : 1c) 3 : 4d) None of theseCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for DIRECTIONS for questions: Select the correct alternative from the given choices.Consider two cubes of equal volume. In one of the cubes, the biggest possible sphere is inscribed. The other cube is cut into exactly 64 identical small cubes. In each of them, the biggest possible sphere is placed. The volume of the sphere in the first cube is denoted by V1. The total volume of the spheres in the small cubes is denoted by V2. Find V1 : V2.a) 4 : 3b) 1 : 1c) 3 : 4d) None of theseCorrect answer is option 'B'. Can you explain this answer?.
Solutions for DIRECTIONS for questions: Select the correct alternative from the given choices.Consider two cubes of equal volume. In one of the cubes, the biggest possible sphere is inscribed. The other cube is cut into exactly 64 identical small cubes. In each of them, the biggest possible sphere is placed. The volume of the sphere in the first cube is denoted by V1. The total volume of the spheres in the small cubes is denoted by V2. Find V1 : V2.a) 4 : 3b) 1 : 1c) 3 : 4d) None of theseCorrect answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for CAT.
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DIRECTIONS for questions: Select the correct alternative from the given choices.Consider two cubes of equal volume. In one of the cubes, the biggest possible sphere is inscribed. The other cube is cut into exactly 64 identical small cubes. In each of them, the biggest possible sphere is placed. The volume of the sphere in the first cube is denoted by V1. The total volume of the spheres in the small cubes is denoted by V2. Find V1 : V2.a) 4 : 3b) 1 : 1c) 3 : 4d) None of theseCorrect answer is option 'B'. Can you explain this answer?, a detailed solution for DIRECTIONS for questions: Select the correct alternative from the given choices.Consider two cubes of equal volume. In one of the cubes, the biggest possible sphere is inscribed. The other cube is cut into exactly 64 identical small cubes. In each of them, the biggest possible sphere is placed. The volume of the sphere in the first cube is denoted by V1. The total volume of the spheres in the small cubes is denoted by V2. Find V1 : V2.a) 4 : 3b) 1 : 1c) 3 : 4d) None of theseCorrect answer is option 'B'. Can you explain this answer? has been provided alongside types of DIRECTIONS for questions: Select the correct alternative from the given choices.Consider two cubes of equal volume. In one of the cubes, the biggest possible sphere is inscribed. The other cube is cut into exactly 64 identical small cubes. In each of them, the biggest possible sphere is placed. The volume of the sphere in the first cube is denoted by V1. The total volume of the spheres in the small cubes is denoted by V2. Find V1 : V2.a) 4 : 3b) 1 : 1c) 3 : 4d) None of theseCorrect answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice DIRECTIONS for questions: Select the correct alternative from the given choices.Consider two cubes of equal volume. In one of the cubes, the biggest possible sphere is inscribed. The other cube is cut into exactly 64 identical small cubes. In each of them, the biggest possible sphere is placed. The volume of the sphere in the first cube is denoted by V1. The total volume of the spheres in the small cubes is denoted by V2. Find V1 : V2.a) 4 : 3b) 1 : 1c) 3 : 4d) None of theseCorrect answer is option 'B'. Can you explain this answer? tests, examples and also practice CAT tests.
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