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DIRECTIONS for questions: Select the correct alternative from the given choices.
Five identical tennis balls are kept in a cylindrical container whose length is such that the five balls just fit into it in a single column. Find the ratio of the volume of the unoccupied portion of the container to that of the occupied portion, given that each ball is of the maximum possible size that can fit inside the container.
- a)3 : 1
- b)2 : 1
- c)2 : 3
- d)1 : 2
Correct answer is option 'D'. Can you explain this answer?
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DIRECTIONS for questions: Select the correct alternative from the giv...
The above figure gives the side view of the container and the balls.
Length of the box = 5 x 2r = 10r
Radius of the box = r
Volume of the box = πr2 x 10r = π10r3
Total volume of the five balls = 5 x 4/3 x π x r3 = 20r3π / 3
Unoccupied volume = 10πr3(1 – 2/3) = 10πr3(1 /3).
∴Required ratio = 10πr3: 20/3 πr3 =1:2
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DIRECTIONS for questions: Select the correct alternative from the giv...
Ratio of the volume of the unoccupied portion of the container to that of the occupied portion can be found by comparing the volume of the empty space with the volume occupied by the tennis balls.
Let's assume the radius of each tennis ball is 'r' and the height of the cylindrical container is 'h'.
Volume of a single tennis ball = (4/3)πr^3
Since all the balls are identical, the total volume occupied by the five balls = 5 × (4/3)πr^3 = (20/3)πr^3
Volume of the cylindrical container = πr^2h
To find the ratio, we need to calculate the volume of the unoccupied portion of the container.
Volume of unoccupied portion = Volume of the cylindrical container - Volume of the occupied portion
= (πr^2h) - (20/3)πr^3
= πr^2h - (20/3)πr^3
Now, let's simplify the expression.
Taking π common, we get:
Volume of unoccupied portion = π(r^2h - (20/3)r^3)
To find the ratio, we need to compare the volume of the unoccupied portion with the volume of the occupied portion (5 tennis balls).
Ratio = Volume of unoccupied portion / Volume of occupied portion
= π(r^2h - (20/3)r^3) / (20/3)πr^3
= (r^2h - (20/3)r^3) / ((20/3)r^3)
= (h/r - (20/3)r^2) / (20/3)r^2
Since each ball is of the maximum possible size that can fit inside the container, the radius of each ball will be such that they just fit into the container in a single column. This means the height of the container will be equal to the diameter of a tennis ball, which is 2r.
Substituting h = 2r in the above equation, we get:
Ratio = ((2r)/r - (20/3)r^2) / (20/3)r^2
= (2 - (20/3)r^2) / (20/3)r^2
= (6 - 20r^2) / 20r^2
= 1 - (20r^2 / 20r^2)
= 1 - 1
= 0
Therefore, the ratio of the volume of the unoccupied portion of the container to that of the occupied portion is 0 : 1, which can be simplified to 0 : 1 or 1 : 0.
However, it is important to note that the ratio cannot be determined as the problem assumes that the balls just fit into the container in a single column. In reality, there would be some empty space between the balls which would affect the ratio.
Let's assume the radius of each tennis ball is 'r' and the height of the cylindrical container is 'h'.
Volume of a single tennis ball = (4/3)πr^3
Since all the balls are identical, the total volume occupied by the five balls = 5 × (4/3)πr^3 = (20/3)πr^3
Volume of the cylindrical container = πr^2h
To find the ratio, we need to calculate the volume of the unoccupied portion of the container.
Volume of unoccupied portion = Volume of the cylindrical container - Volume of the occupied portion
= (πr^2h) - (20/3)πr^3
= πr^2h - (20/3)πr^3
Now, let's simplify the expression.
Taking π common, we get:
Volume of unoccupied portion = π(r^2h - (20/3)r^3)
To find the ratio, we need to compare the volume of the unoccupied portion with the volume of the occupied portion (5 tennis balls).
Ratio = Volume of unoccupied portion / Volume of occupied portion
= π(r^2h - (20/3)r^3) / (20/3)πr^3
= (r^2h - (20/3)r^3) / ((20/3)r^3)
= (h/r - (20/3)r^2) / (20/3)r^2
Since each ball is of the maximum possible size that can fit inside the container, the radius of each ball will be such that they just fit into the container in a single column. This means the height of the container will be equal to the diameter of a tennis ball, which is 2r.
Substituting h = 2r in the above equation, we get:
Ratio = ((2r)/r - (20/3)r^2) / (20/3)r^2
= (2 - (20/3)r^2) / (20/3)r^2
= (6 - 20r^2) / 20r^2
= 1 - (20r^2 / 20r^2)
= 1 - 1
= 0
Therefore, the ratio of the volume of the unoccupied portion of the container to that of the occupied portion is 0 : 1, which can be simplified to 0 : 1 or 1 : 0.
However, it is important to note that the ratio cannot be determined as the problem assumes that the balls just fit into the container in a single column. In reality, there would be some empty space between the balls which would affect the ratio.
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DIRECTIONS for questions: Select the correct alternative from the given choices. Five identical tennis balls are kept in a cylindrical container whose length is such that the five balls just fit into it in a single column. Find the ratio of the volume of the unoccupied portion of the container to that of the occupied portion, given that each ball is of the maximum possible size that can fit inside the container.a) 3 : 1b) 2 : 1c) 2 : 3d) 1 : 2Correct answer is option 'D'. Can you explain this answer?
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DIRECTIONS for questions: Select the correct alternative from the given choices. Five identical tennis balls are kept in a cylindrical container whose length is such that the five balls just fit into it in a single column. Find the ratio of the volume of the unoccupied portion of the container to that of the occupied portion, given that each ball is of the maximum possible size that can fit inside the container.a) 3 : 1b) 2 : 1c) 2 : 3d) 1 : 2Correct answer is option 'D'. 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. Five identical tennis balls are kept in a cylindrical container whose length is such that the five balls just fit into it in a single column. Find the ratio of the volume of the unoccupied portion of the container to that of the occupied portion, given that each ball is of the maximum possible size that can fit inside the container.a) 3 : 1b) 2 : 1c) 2 : 3d) 1 : 2Correct answer is option 'D'. 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. Five identical tennis balls are kept in a cylindrical container whose length is such that the five balls just fit into it in a single column. Find the ratio of the volume of the unoccupied portion of the container to that of the occupied portion, given that each ball is of the maximum possible size that can fit inside the container.a) 3 : 1b) 2 : 1c) 2 : 3d) 1 : 2Correct answer is option 'D'. Can you explain this answer?.
DIRECTIONS for questions: Select the correct alternative from the given choices. Five identical tennis balls are kept in a cylindrical container whose length is such that the five balls just fit into it in a single column. Find the ratio of the volume of the unoccupied portion of the container to that of the occupied portion, given that each ball is of the maximum possible size that can fit inside the container.a) 3 : 1b) 2 : 1c) 2 : 3d) 1 : 2Correct answer is option 'D'. 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. Five identical tennis balls are kept in a cylindrical container whose length is such that the five balls just fit into it in a single column. Find the ratio of the volume of the unoccupied portion of the container to that of the occupied portion, given that each ball is of the maximum possible size that can fit inside the container.a) 3 : 1b) 2 : 1c) 2 : 3d) 1 : 2Correct answer is option 'D'. 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. Five identical tennis balls are kept in a cylindrical container whose length is such that the five balls just fit into it in a single column. Find the ratio of the volume of the unoccupied portion of the container to that of the occupied portion, given that each ball is of the maximum possible size that can fit inside the container.a) 3 : 1b) 2 : 1c) 2 : 3d) 1 : 2Correct answer is option 'D'. Can you explain this answer?.
Solutions for DIRECTIONS for questions: Select the correct alternative from the given choices. Five identical tennis balls are kept in a cylindrical container whose length is such that the five balls just fit into it in a single column. Find the ratio of the volume of the unoccupied portion of the container to that of the occupied portion, given that each ball is of the maximum possible size that can fit inside the container.a) 3 : 1b) 2 : 1c) 2 : 3d) 1 : 2Correct answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for CAT.
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