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Let S1 be a square of side a units. A circle C1 is inscribed in S1. Another square, S2, is inscribed inside the circle C1. Another circle, C2, is inscribed inside the square S2. Another square, S3, is inscribed inside the circle C2 and so on. Find the ratio of perimeters of all the circles to perimeters of all the squares.
- a)4/π
- b)4-π/2
- c)π-2/2
- d)None of the above
Correct answer is option 'D'. Can you explain this answer?
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Let S1 be a square of side a units. A circle C1 is inscribed in S1. An...
Hence, option 4.
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Let S1 be a square of side a units. A circle C1 is inscribed in S1. An...
Let's start by finding the relationship between the side lengths of the squares and the radii of the circles.
Let S1 be the first square with side length a.
The circle C1 inscribed in S1 has a radius equal to half the side length of S1, which is a/2.
The second square, S2, inscribed in C1, has a side length equal to the diameter of C1, which is twice the radius, or a.
The circle C2 inscribed in S2 has a radius equal to half the side length of S2, which is a/2.
The third square, S3, inscribed in C2, has a side length equal to the diameter of C2, which is twice the radius, or a.
We can see that for each iteration, the side length of the square is equal to the diameter of the previous circle, and the diameter of the circle is equal to the side length of the previous square.
Therefore, the ratio of the side length of each square to the radius of each circle is always 2:1.
Now let's consider the perimeters. The perimeter of a square is given by 4 times the side length, and the perimeter of a circle is given by 2 times pi times the radius.
Let's denote the perimeters of the squares as P1, P2, P3,... and the perimeters of the circles as C1, C2, C3,...
We can see that the ratio of the perimeters of the squares is given by:
P1 : P2 = a : 2a = 1 : 2
P2 : P3 = a : 2a = 1 : 2
P3 : P4 = a : 2a = 1 : 2
Therefore, the ratio of the perimeters of all the squares is 1 : 2 : 2 : 2 : ...
Similarly, the ratio of the perimeters of the circles is given by:
C1 : C2 = 2(pi)(a/2) : 2(pi)(a/4) = pi : pi/2 = 2 : 1
C2 : C3 = 2(pi)(a/4) : 2(pi)(a/8) = pi/2 : pi/4 = 2 : 1
C3 : C4 = 2(pi)(a/8) : 2(pi)(a/16) = pi/4 : pi/8 = 2 : 1
Therefore, the ratio of the perimeters of all the circles is 2 : 1 : 1 : 1 : ...
Combining the ratios of the perimeters of the squares and the circles, we have:
Perimeters of all the squares : Perimeters of all the circles = 1 : 2 : 2 : 2 : ... : 2 : 1 : 1 : 1 : ...
This can be simplified to:
Perimeters of all the squares : Perimeters of all the circles = 1 : 2
Therefore, the ratio of the perimeters of all the circles to the perimeters of all the squares is 2/1, or simply 2.
So the answer is:
a) 2
Let S1 be the first square with side length a.
The circle C1 inscribed in S1 has a radius equal to half the side length of S1, which is a/2.
The second square, S2, inscribed in C1, has a side length equal to the diameter of C1, which is twice the radius, or a.
The circle C2 inscribed in S2 has a radius equal to half the side length of S2, which is a/2.
The third square, S3, inscribed in C2, has a side length equal to the diameter of C2, which is twice the radius, or a.
We can see that for each iteration, the side length of the square is equal to the diameter of the previous circle, and the diameter of the circle is equal to the side length of the previous square.
Therefore, the ratio of the side length of each square to the radius of each circle is always 2:1.
Now let's consider the perimeters. The perimeter of a square is given by 4 times the side length, and the perimeter of a circle is given by 2 times pi times the radius.
Let's denote the perimeters of the squares as P1, P2, P3,... and the perimeters of the circles as C1, C2, C3,...
We can see that the ratio of the perimeters of the squares is given by:
P1 : P2 = a : 2a = 1 : 2
P2 : P3 = a : 2a = 1 : 2
P3 : P4 = a : 2a = 1 : 2
Therefore, the ratio of the perimeters of all the squares is 1 : 2 : 2 : 2 : ...
Similarly, the ratio of the perimeters of the circles is given by:
C1 : C2 = 2(pi)(a/2) : 2(pi)(a/4) = pi : pi/2 = 2 : 1
C2 : C3 = 2(pi)(a/4) : 2(pi)(a/8) = pi/2 : pi/4 = 2 : 1
C3 : C4 = 2(pi)(a/8) : 2(pi)(a/16) = pi/4 : pi/8 = 2 : 1
Therefore, the ratio of the perimeters of all the circles is 2 : 1 : 1 : 1 : ...
Combining the ratios of the perimeters of the squares and the circles, we have:
Perimeters of all the squares : Perimeters of all the circles = 1 : 2 : 2 : 2 : ... : 2 : 1 : 1 : 1 : ...
This can be simplified to:
Perimeters of all the squares : Perimeters of all the circles = 1 : 2
Therefore, the ratio of the perimeters of all the circles to the perimeters of all the squares is 2/1, or simply 2.
So the answer is:
a) 2
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Let S1 be a square of side a units. A circle C1 is inscribed in S1. Another square, S2, is inscribed inside the circle C1. Another circle, C2, is inscribed inside the square S2. Another square, S3, is inscribed inside the circle C2 and so on. Find the ratio of perimeters of all the circles to perimeters of all the squares.a)4/πb)4-π/2c)π-2/2d)None of the aboveCorrect answer is option 'D'. Can you explain this answer?
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Let S1 be a square of side a units. A circle C1 is inscribed in S1. Another square, S2, is inscribed inside the circle C1. Another circle, C2, is inscribed inside the square S2. Another square, S3, is inscribed inside the circle C2 and so on. Find the ratio of perimeters of all the circles to perimeters of all the squares.a)4/πb)4-π/2c)π-2/2d)None of the aboveCorrect 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 Let S1 be a square of side a units. A circle C1 is inscribed in S1. Another square, S2, is inscribed inside the circle C1. Another circle, C2, is inscribed inside the square S2. Another square, S3, is inscribed inside the circle C2 and so on. Find the ratio of perimeters of all the circles to perimeters of all the squares.a)4/πb)4-π/2c)π-2/2d)None of the aboveCorrect 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 Let S1 be a square of side a units. A circle C1 is inscribed in S1. Another square, S2, is inscribed inside the circle C1. Another circle, C2, is inscribed inside the square S2. Another square, S3, is inscribed inside the circle C2 and so on. Find the ratio of perimeters of all the circles to perimeters of all the squares.a)4/πb)4-π/2c)π-2/2d)None of the aboveCorrect answer is option 'D'. Can you explain this answer?.
Let S1 be a square of side a units. A circle C1 is inscribed in S1. Another square, S2, is inscribed inside the circle C1. Another circle, C2, is inscribed inside the square S2. Another square, S3, is inscribed inside the circle C2 and so on. Find the ratio of perimeters of all the circles to perimeters of all the squares.a)4/πb)4-π/2c)π-2/2d)None of the aboveCorrect 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 Let S1 be a square of side a units. A circle C1 is inscribed in S1. Another square, S2, is inscribed inside the circle C1. Another circle, C2, is inscribed inside the square S2. Another square, S3, is inscribed inside the circle C2 and so on. Find the ratio of perimeters of all the circles to perimeters of all the squares.a)4/πb)4-π/2c)π-2/2d)None of the aboveCorrect 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 Let S1 be a square of side a units. A circle C1 is inscribed in S1. Another square, S2, is inscribed inside the circle C1. Another circle, C2, is inscribed inside the square S2. Another square, S3, is inscribed inside the circle C2 and so on. Find the ratio of perimeters of all the circles to perimeters of all the squares.a)4/πb)4-π/2c)π-2/2d)None of the aboveCorrect answer is option 'D'. Can you explain this answer?.
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Let S1 be a square of side a units. A circle C1 is inscribed in S1. Another square, S2, is inscribed inside the circle C1. Another circle, C2, is inscribed inside the square S2. Another square, S3, is inscribed inside the circle C2 and so on. Find the ratio of perimeters of all the circles to perimeters of all the squares.a)4/πb)4-π/2c)π-2/2d)None of the aboveCorrect answer is option 'D'. Can you explain this answer?, a detailed solution for Let S1 be a square of side a units. A circle C1 is inscribed in S1. Another square, S2, is inscribed inside the circle C1. Another circle, C2, is inscribed inside the square S2. Another square, S3, is inscribed inside the circle C2 and so on. Find the ratio of perimeters of all the circles to perimeters of all the squares.a)4/πb)4-π/2c)π-2/2d)None of the aboveCorrect answer is option 'D'. Can you explain this answer? has been provided alongside types of Let S1 be a square of side a units. A circle C1 is inscribed in S1. Another square, S2, is inscribed inside the circle C1. Another circle, C2, is inscribed inside the square S2. Another square, S3, is inscribed inside the circle C2 and so on. Find the ratio of perimeters of all the circles to perimeters of all the squares.a)4/πb)4-π/2c)π-2/2d)None of the aboveCorrect answer is option 'D'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Let S1 be a square of side a units. A circle C1 is inscribed in S1. Another square, S2, is inscribed inside the circle C1. Another circle, C2, is inscribed inside the square S2. Another square, S3, is inscribed inside the circle C2 and so on. Find the ratio of perimeters of all the circles to perimeters of all the squares.a)4/πb)4-π/2c)π-2/2d)None of the aboveCorrect answer is option 'D'. Can you explain this answer? tests, examples and also practice CAT tests.
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