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Consider an infinite number of cylinders. The first cylinder has a radius of 1 meter and height of 1 meter. The second one has a radius of 0.5 meter and height of 0.5 meter. Every subsequent cylinder has half the radius and half the height of the preceding cylinder. The sum of the volumes (in cubic meters) of these infinite number of cylinders is ____.
Given data: π = 3.14
Correct answer is '3.62'. Can you explain this answer?
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Consider an infinite number of cylinders. The first cylinder has a rad...
The given problem involves an infinite number of cylinders, each with a progressively smaller radius and height. We need to find the sum of the volumes of all these cylinders.
To solve this problem, we can start by finding the volume of the first cylinder.
Volume of the first cylinder:
The first cylinder has a radius of 1 meter and a height of 1 meter. The formula to find the volume of a cylinder is V = πr²h, where V is the volume, r is the radius, and h is the height.
Using the given data, we can calculate the volume of the first cylinder as:
V₁ = π(1²)(1) = 3.14 cubic meters.
Volume of the second cylinder:
The second cylinder has half the radius and half the height of the first cylinder. Therefore, the radius of the second cylinder is 0.5 meters and the height is 0.5 meters.
Using the formula for the volume of a cylinder, we can find the volume of the second cylinder as:
V₂ = π(0.5²)(0.5) = 0.785 cubic meters.
Volume of the third cylinder:
Following the same pattern, the radius of the third cylinder is 0.25 meters and the height is 0.25 meters.
Using the volume formula, we can find the volume of the third cylinder as:
V₃ = π(0.25²)(0.25) = 0.196 cubic meters.
Pattern and General Formula:
From the calculations above, we can observe a pattern emerging. The volume of each cylinder is one-eighth of the volume of the preceding cylinder. Therefore, we can establish a general formula to find the volume of any cylinder in the sequence.
The volume of the nth cylinder can be calculated using the formula:
Vₙ = π(1/2)^(2n-2)(1/2)^(2n-2) = π(1/2)^(4n-4).
Sum of the volumes:
Now, we need to find the sum of the volumes of all the cylinders. Since we have an infinite number of cylinders, we need to use the concept of an infinite geometric series.
The sum of an infinite geometric series can be calculated using the formula:
S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio.
In our case, the first term (a) is V₁ = 3.14 cubic meters, and the common ratio (r) is 1/8.
Using the formula, we can calculate the sum of the volumes as:
S = 3.14 / (1 - 1/8) = 3.14 / (7/8) = 3.62 cubic meters.
Therefore, the sum of the volumes of the infinite number of cylinders is 3.62 cubic meters.
To solve this problem, we can start by finding the volume of the first cylinder.
Volume of the first cylinder:
The first cylinder has a radius of 1 meter and a height of 1 meter. The formula to find the volume of a cylinder is V = πr²h, where V is the volume, r is the radius, and h is the height.
Using the given data, we can calculate the volume of the first cylinder as:
V₁ = π(1²)(1) = 3.14 cubic meters.
Volume of the second cylinder:
The second cylinder has half the radius and half the height of the first cylinder. Therefore, the radius of the second cylinder is 0.5 meters and the height is 0.5 meters.
Using the formula for the volume of a cylinder, we can find the volume of the second cylinder as:
V₂ = π(0.5²)(0.5) = 0.785 cubic meters.
Volume of the third cylinder:
Following the same pattern, the radius of the third cylinder is 0.25 meters and the height is 0.25 meters.
Using the volume formula, we can find the volume of the third cylinder as:
V₃ = π(0.25²)(0.25) = 0.196 cubic meters.
Pattern and General Formula:
From the calculations above, we can observe a pattern emerging. The volume of each cylinder is one-eighth of the volume of the preceding cylinder. Therefore, we can establish a general formula to find the volume of any cylinder in the sequence.
The volume of the nth cylinder can be calculated using the formula:
Vₙ = π(1/2)^(2n-2)(1/2)^(2n-2) = π(1/2)^(4n-4).
Sum of the volumes:
Now, we need to find the sum of the volumes of all the cylinders. Since we have an infinite number of cylinders, we need to use the concept of an infinite geometric series.
The sum of an infinite geometric series can be calculated using the formula:
S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio.
In our case, the first term (a) is V₁ = 3.14 cubic meters, and the common ratio (r) is 1/8.
Using the formula, we can calculate the sum of the volumes as:
S = 3.14 / (1 - 1/8) = 3.14 / (7/8) = 3.62 cubic meters.
Therefore, the sum of the volumes of the infinite number of cylinders is 3.62 cubic meters.
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Consider an infinite number of cylinders. The first cylinder has a radius of 1 meter and height of 1 meter. The second one has a radius of 0.5 meter and height of 0.5 meter. Every subsequent cylinder has half the radius and half the height of the preceding cylinder. The sum of the volumes (in cubic meters) of these infinite number of cylinders is ____.Given data: π = 3.14Correct answer is '3.62'. Can you explain this answer?
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Consider an infinite number of cylinders. The first cylinder has a radius of 1 meter and height of 1 meter. The second one has a radius of 0.5 meter and height of 0.5 meter. Every subsequent cylinder has half the radius and half the height of the preceding cylinder. The sum of the volumes (in cubic meters) of these infinite number of cylinders is ____.Given data: π = 3.14Correct answer is '3.62'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Consider an infinite number of cylinders. The first cylinder has a radius of 1 meter and height of 1 meter. The second one has a radius of 0.5 meter and height of 0.5 meter. Every subsequent cylinder has half the radius and half the height of the preceding cylinder. The sum of the volumes (in cubic meters) of these infinite number of cylinders is ____.Given data: π = 3.14Correct answer is '3.62'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider an infinite number of cylinders. The first cylinder has a radius of 1 meter and height of 1 meter. The second one has a radius of 0.5 meter and height of 0.5 meter. Every subsequent cylinder has half the radius and half the height of the preceding cylinder. The sum of the volumes (in cubic meters) of these infinite number of cylinders is ____.Given data: π = 3.14Correct answer is '3.62'. Can you explain this answer?.
Consider an infinite number of cylinders. The first cylinder has a radius of 1 meter and height of 1 meter. The second one has a radius of 0.5 meter and height of 0.5 meter. Every subsequent cylinder has half the radius and half the height of the preceding cylinder. The sum of the volumes (in cubic meters) of these infinite number of cylinders is ____.Given data: π = 3.14Correct answer is '3.62'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Consider an infinite number of cylinders. The first cylinder has a radius of 1 meter and height of 1 meter. The second one has a radius of 0.5 meter and height of 0.5 meter. Every subsequent cylinder has half the radius and half the height of the preceding cylinder. The sum of the volumes (in cubic meters) of these infinite number of cylinders is ____.Given data: π = 3.14Correct answer is '3.62'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider an infinite number of cylinders. The first cylinder has a radius of 1 meter and height of 1 meter. The second one has a radius of 0.5 meter and height of 0.5 meter. Every subsequent cylinder has half the radius and half the height of the preceding cylinder. The sum of the volumes (in cubic meters) of these infinite number of cylinders is ____.Given data: π = 3.14Correct answer is '3.62'. Can you explain this answer?.
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