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A cylindircal gas container is closed at the top and open at the bottom. if the iron plate of the top is 5/4 time as thick as the plate forming thecylindrical sides. The ratio of the radius to the height of the cylinder using minimum material for the same capacity is
- a)2/3
- b)1/2
- c)4/5
- d)1/3
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
A cylindircal gas container is closed at the top and open at the botto...
V = πr2h = constant. If k be the thickness of the sides then that of the top will be (5/4)k.
∴ S = (2πrh)k + (πr2). (5/4)k
(‘S’ is vol. of material used)
When r3 = 4V/5π or 5πr3 = 4πr2h.
∴ r/h = 5/4
∴ S = (2πrh)k + (πr2). (5/4)k
(‘S’ is vol. of material used)
When r3 = 4V/5π or 5πr3 = 4πr2h.
∴ r/h = 5/4
Most Upvoted Answer
A cylindircal gas container is closed at the top and open at the botto...
Let's assume the thickness of the plate forming the cylindrical sides of the gas container is 't' (in any unit), and the height of the cylinder is 'h' (in the same unit).
We are given that the iron plate on the top is 5/4 times thicker than the plate forming the cylindrical sides. Therefore, the thickness of the top plate is (5/4)t.
To find the ratio of the radius to the height of the cylinder using the minimum material for the same capacity, we need to minimize the surface area of the container while keeping the volume constant.
We know that the volume of a cylinder is given by the formula:
V = πr^2h
To keep the volume constant, we need to keep the value of V constant. Therefore, we can write:
πr^2h = constant
Let's calculate the surface area of the cylinder, which is the sum of the curved surface area and the top area.
1. Curved Surface Area:
The curved surface area of a cylinder is given by the formula:
CSA = 2πrh
2. Top Area:
The top area of the cylinder is the area of the circular iron plate, which is given by the formula:
A = πr^2
Now, let's calculate the total surface area (TSA) of the gas container by summing up the curved surface area and the top area:
TSA = CSA + A
= 2πrh + πr^2
= πr(2h + r)
To minimize the surface area, we need to minimize the function TSA = πr(2h + r) while keeping the volume constant.
Let's differentiate the TSA function with respect to 'r' and set it equal to zero to find the critical points:
d(TSA)/dr = 2πh + 2πr = 0
2πh + 2πr = 0
r = -h
Since 'r' cannot be negative, we discard this critical point.
Therefore, the critical points do not exist, which means that the minimum value of the surface area occurs at the endpoints of the interval.
Since we are looking for the minimum value, we need to consider the smallest possible value for 'r'. Since 'r' cannot be zero (as it would result in a flat plate), we consider the next smallest value, which is (5/4)t.
Therefore, the ratio of the radius to the height of the cylinder using the minimum material for the same capacity is:
(r/h) = [(5/4)t/h] = (5/4)
Simplifying this ratio, we get:
(r/h) = 5/4 = 4/5
Hence, the correct answer is option 'C' (4/5).
We are given that the iron plate on the top is 5/4 times thicker than the plate forming the cylindrical sides. Therefore, the thickness of the top plate is (5/4)t.
To find the ratio of the radius to the height of the cylinder using the minimum material for the same capacity, we need to minimize the surface area of the container while keeping the volume constant.
We know that the volume of a cylinder is given by the formula:
V = πr^2h
To keep the volume constant, we need to keep the value of V constant. Therefore, we can write:
πr^2h = constant
Let's calculate the surface area of the cylinder, which is the sum of the curved surface area and the top area.
1. Curved Surface Area:
The curved surface area of a cylinder is given by the formula:
CSA = 2πrh
2. Top Area:
The top area of the cylinder is the area of the circular iron plate, which is given by the formula:
A = πr^2
Now, let's calculate the total surface area (TSA) of the gas container by summing up the curved surface area and the top area:
TSA = CSA + A
= 2πrh + πr^2
= πr(2h + r)
To minimize the surface area, we need to minimize the function TSA = πr(2h + r) while keeping the volume constant.
Let's differentiate the TSA function with respect to 'r' and set it equal to zero to find the critical points:
d(TSA)/dr = 2πh + 2πr = 0
2πh + 2πr = 0
r = -h
Since 'r' cannot be negative, we discard this critical point.
Therefore, the critical points do not exist, which means that the minimum value of the surface area occurs at the endpoints of the interval.
Since we are looking for the minimum value, we need to consider the smallest possible value for 'r'. Since 'r' cannot be zero (as it would result in a flat plate), we consider the next smallest value, which is (5/4)t.
Therefore, the ratio of the radius to the height of the cylinder using the minimum material for the same capacity is:
(r/h) = [(5/4)t/h] = (5/4)
Simplifying this ratio, we get:
(r/h) = 5/4 = 4/5
Hence, the correct answer is option 'C' (4/5).
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A cylindircal gas container is closed at the top and open at the bottom. if the iron plate of the top is 5/4 timeas thick as the plate forming thecylindrical sides. The ratio of the radius to the height of the cylinder using minimum material for the same capacity isa)2/3b)1/2c)4/5d)1/3Correct answer is option 'C'. Can you explain this answer?
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A cylindircal gas container is closed at the top and open at the bottom. if the iron plate of the top is 5/4 timeas thick as the plate forming thecylindrical sides. The ratio of the radius to the height of the cylinder using minimum material for the same capacity isa)2/3b)1/2c)4/5d)1/3Correct answer is option 'C'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A cylindircal gas container is closed at the top and open at the bottom. if the iron plate of the top is 5/4 timeas thick as the plate forming thecylindrical sides. The ratio of the radius to the height of the cylinder using minimum material for the same capacity isa)2/3b)1/2c)4/5d)1/3Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A cylindircal gas container is closed at the top and open at the bottom. if the iron plate of the top is 5/4 timeas thick as the plate forming thecylindrical sides. The ratio of the radius to the height of the cylinder using minimum material for the same capacity isa)2/3b)1/2c)4/5d)1/3Correct answer is option 'C'. Can you explain this answer?.
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