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Two castings of the same metal have the same surface area. One castings is in the form of a sphere and the other is a cube. What is the ratio of the solidification time for the sphere to that of the cube?
- a)3/4
- b)6/π
- c)1π
- d)3/π
Correct answer is option 'B'. Can you explain this answer?
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Two castings of the same metal have the same surface area. One castin...
As surface area are equal 4πr2 = 6a2
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Two castings of the same metal have the same surface area. One castin...
Surface Area of a Sphere:
The surface area of a sphere is given by the formula 4πr^2, where r is the radius of the sphere.
Surface Area of a Cube:
The surface area of a cube is given by the formula 6s^2, where s is the length of one side of the cube.
Given:
The two castings have the same surface area.
Let's assume:
Let's assume that the surface area of both the sphere and the cube is equal to A.
Equating the Surface Areas:
4πr^2 = 6s^2
Simplifying the Equation:
Dividing both sides by 2, we get:
2πr^2 = 3s^2
Ratio of Solidification Time:
The solidification time for a casting is directly proportional to its volume. The volume of a sphere is given by the formula (4/3)πr^3, and the volume of a cube is given by the formula s^3.
Let's assume:
Let's assume that the solidification time for the sphere is T1 and the solidification time for the cube is T2.
Equating the Volumes:
(4/3)πr^3 = s^3
Simplifying the Equation:
Dividing both sides by π:
(4/3)r^3 = (s^3)/π
Ratio of Solidification Time:
Since the solidification time is directly proportional to the volume, we can write:
T1/T2 = (4/3)r^3 / (s^3)/π
Substituting the Value of s^2:
Using the equation 2πr^2 = 3s^2, we can substitute the value of s^2 in the ratio equation:
T1/T2 = (4/3)r^3 / (2πr^2)^(3/2)
Simplifying the Equation:
T1/T2 = (4/3) / (2π)^(3/2) * (r^3 / r^3)
T1/T2 = (4/3) / (2π)^(3/2)
Approximating the Value:
Using the approximation π ≈ 3.14, we can simplify the equation further:
T1/T2 ≈ (4/3) / (2 * 3.14)^(3/2)
Calculating the Value:
T1/T2 ≈ (4/3) / (2 * 3.14)^(3/2)
T1/T2 ≈ (4/3) / (2 * 5.61)
T1/T2 ≈ 0.238
Final Answer:
The ratio of the solidification time for the sphere to that of the cube is approximately 0.238, which is equivalent to 6/π. Therefore, the correct answer is option B.
The surface area of a sphere is given by the formula 4πr^2, where r is the radius of the sphere.
Surface Area of a Cube:
The surface area of a cube is given by the formula 6s^2, where s is the length of one side of the cube.
Given:
The two castings have the same surface area.
Let's assume:
Let's assume that the surface area of both the sphere and the cube is equal to A.
Equating the Surface Areas:
4πr^2 = 6s^2
Simplifying the Equation:
Dividing both sides by 2, we get:
2πr^2 = 3s^2
Ratio of Solidification Time:
The solidification time for a casting is directly proportional to its volume. The volume of a sphere is given by the formula (4/3)πr^3, and the volume of a cube is given by the formula s^3.
Let's assume:
Let's assume that the solidification time for the sphere is T1 and the solidification time for the cube is T2.
Equating the Volumes:
(4/3)πr^3 = s^3
Simplifying the Equation:
Dividing both sides by π:
(4/3)r^3 = (s^3)/π
Ratio of Solidification Time:
Since the solidification time is directly proportional to the volume, we can write:
T1/T2 = (4/3)r^3 / (s^3)/π
Substituting the Value of s^2:
Using the equation 2πr^2 = 3s^2, we can substitute the value of s^2 in the ratio equation:
T1/T2 = (4/3)r^3 / (2πr^2)^(3/2)
Simplifying the Equation:
T1/T2 = (4/3) / (2π)^(3/2) * (r^3 / r^3)
T1/T2 = (4/3) / (2π)^(3/2)
Approximating the Value:
Using the approximation π ≈ 3.14, we can simplify the equation further:
T1/T2 ≈ (4/3) / (2 * 3.14)^(3/2)
Calculating the Value:
T1/T2 ≈ (4/3) / (2 * 3.14)^(3/2)
T1/T2 ≈ (4/3) / (2 * 5.61)
T1/T2 ≈ 0.238
Final Answer:
The ratio of the solidification time for the sphere to that of the cube is approximately 0.238, which is equivalent to 6/π. Therefore, the correct answer is option B.
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Two castings of the same metal have the same surface area. One castings is in the form of a sphere and the other is a cube. What is the ratio of the solidification time for the sphere to that of the cube?a)3/4b)6/πc)1πd)3/πCorrect answer is option 'B'. Can you explain this answer?
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Two castings of the same metal have the same surface area. One castings is in the form of a sphere and the other is a cube. What is the ratio of the solidification time for the sphere to that of the cube?a)3/4b)6/πc)1πd)3/πCorrect answer is option 'B'. 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 Two castings of the same metal have the same surface area. One castings is in the form of a sphere and the other is a cube. What is the ratio of the solidification time for the sphere to that of the cube?a)3/4b)6/πc)1πd)3/πCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two castings of the same metal have the same surface area. One castings is in the form of a sphere and the other is a cube. What is the ratio of the solidification time for the sphere to that of the cube?a)3/4b)6/πc)1πd)3/πCorrect answer is option 'B'. Can you explain this answer?.
Two castings of the same metal have the same surface area. One castings is in the form of a sphere and the other is a cube. What is the ratio of the solidification time for the sphere to that of the cube?a)3/4b)6/πc)1πd)3/πCorrect answer is option 'B'. 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 Two castings of the same metal have the same surface area. One castings is in the form of a sphere and the other is a cube. What is the ratio of the solidification time for the sphere to that of the cube?a)3/4b)6/πc)1πd)3/πCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two castings of the same metal have the same surface area. One castings is in the form of a sphere and the other is a cube. What is the ratio of the solidification time for the sphere to that of the cube?a)3/4b)6/πc)1πd)3/πCorrect answer is option 'B'. Can you explain this answer?.
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