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There are two identical cubes (C1 and C2 ). C1 is perfectly cut into N identical small cubes. A sphere is inscribed in each of these cubes, such that it occupies maximum volume of the cube. Similarly, A sphere is also inscribed in C2 . The total volume occupied by the spheres in C1 is V 1 and that o f C2 is V2. Find V1 : V2.
- a)1 : 1
- b)1 : 2
- c)2 : 1
- d)4 : 3
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
There are two identical cubes (C1and C2 ). C1 is perfectly cut into N ...
Let the side of a smaller cube is s and that of the bigger cube is b.
s3 x N =b3
s3 x N =b3
Hence, option 1.
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Most Upvoted Answer
There are two identical cubes (C1and C2 ). C1 is perfectly cut into N ...
To find the ratio of the volumes occupied by the spheres in C1 and C2, we need to understand the relationship between the number of small cubes and the total volume occupied by the spheres in each cube.
Let's consider C1, which is perfectly cut into N identical small cubes. Since the sphere occupies maximum volume of each cube, we can assume that the sphere is inscribed in each small cube.
1. Understanding the Volume of a Sphere:
The volume of a sphere is given by the formula V = (4/3)πr³, where r is the radius of the sphere.
2. Relationship between the Number of Small Cubes and the Volume of the Sphere:
In C1, each small cube has the same side length. Let's assume this side length as 'a'. Since the sphere is inscribed in each small cube, the diameter of the sphere is equal to the side length of the small cube, which is 'a'. Therefore, the radius of the sphere is (a/2).
The volume of the sphere inscribed in each small cube is then given by V1 = (4/3)π(a/2)³ = (1/6)πa³.
3. Relationship between the Number of Small Cubes and the Total Volume:
Since C1 is perfectly cut into N identical small cubes, the total volume of C1 is given by V_total = N * V1 = N * (1/6)πa³.
4. Relationship between C1 and C2:
Now, let's consider C2, which is identical to C1. This means that the dimensions of C2 are the same as C1, and it is also perfectly cut into N identical small cubes.
Therefore, the total volume of C2 is also given by V_total = N * (1/6)πa³.
5. Finding the Ratio of V1 to V2:
Since the total volume occupied by the spheres in C1 and C2 is the same, we can conclude that V1 = V2.
Hence, the ratio of V1 to V2 is 1:1, which corresponds to option 'A'.
Let's consider C1, which is perfectly cut into N identical small cubes. Since the sphere occupies maximum volume of each cube, we can assume that the sphere is inscribed in each small cube.
1. Understanding the Volume of a Sphere:
The volume of a sphere is given by the formula V = (4/3)πr³, where r is the radius of the sphere.
2. Relationship between the Number of Small Cubes and the Volume of the Sphere:
In C1, each small cube has the same side length. Let's assume this side length as 'a'. Since the sphere is inscribed in each small cube, the diameter of the sphere is equal to the side length of the small cube, which is 'a'. Therefore, the radius of the sphere is (a/2).
The volume of the sphere inscribed in each small cube is then given by V1 = (4/3)π(a/2)³ = (1/6)πa³.
3. Relationship between the Number of Small Cubes and the Total Volume:
Since C1 is perfectly cut into N identical small cubes, the total volume of C1 is given by V_total = N * V1 = N * (1/6)πa³.
4. Relationship between C1 and C2:
Now, let's consider C2, which is identical to C1. This means that the dimensions of C2 are the same as C1, and it is also perfectly cut into N identical small cubes.
Therefore, the total volume of C2 is also given by V_total = N * (1/6)πa³.
5. Finding the Ratio of V1 to V2:
Since the total volume occupied by the spheres in C1 and C2 is the same, we can conclude that V1 = V2.
Hence, the ratio of V1 to V2 is 1:1, which corresponds to option 'A'.
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There are two identical cubes (C1and C2 ). C1 is perfectly cut into N identical small cubes. A sphere is inscribed in each of these cubes, such that it occupies maximum volume of the cube. Similarly, A sphere is also inscribed in C2 . The total volume occupied by the spheres in C1is V 1 and that o f C2 is V2. Find V1 : V2.a)1 : 1b)1 : 2c)2 : 1d)4 : 3Correct answer is option 'A'. Can you explain this answer?
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There are two identical cubes (C1and C2 ). C1 is perfectly cut into N identical small cubes. A sphere is inscribed in each of these cubes, such that it occupies maximum volume of the cube. Similarly, A sphere is also inscribed in C2 . The total volume occupied by the spheres in C1is V 1 and that o f C2 is V2. Find V1 : V2.a)1 : 1b)1 : 2c)2 : 1d)4 : 3Correct answer is option 'A'. 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 There are two identical cubes (C1and C2 ). C1 is perfectly cut into N identical small cubes. A sphere is inscribed in each of these cubes, such that it occupies maximum volume of the cube. Similarly, A sphere is also inscribed in C2 . The total volume occupied by the spheres in C1is V 1 and that o f C2 is V2. Find V1 : V2.a)1 : 1b)1 : 2c)2 : 1d)4 : 3Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for There are two identical cubes (C1and C2 ). C1 is perfectly cut into N identical small cubes. A sphere is inscribed in each of these cubes, such that it occupies maximum volume of the cube. Similarly, A sphere is also inscribed in C2 . The total volume occupied by the spheres in C1is V 1 and that o f C2 is V2. Find V1 : V2.a)1 : 1b)1 : 2c)2 : 1d)4 : 3Correct answer is option 'A'. Can you explain this answer?.
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