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A cylindrical gas container is closed at the top and open at the bottom. If the iron plate of the top is 5/4 times as thick as the plate forming the cylindrical 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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A cylindrical gas container is closed at the top and open at the botto...
Given: The iron plate of the top is 5/4 times as thick as the plate forming the cylindrical sides.
Let the thickness of the plate forming the cylindrical sides be x.
Then, the thickness of the iron plate of the top will be 5/4x.
Let the radius of the cylinder be r and the height of the cylinder be h.
Volume of the cylinder = πr^2h
The minimum material required to make the cylinder will be when the total surface area is minimum.
Total surface area of the cylinder = Area of top + Area of bottom + Curved surface area
Area of top = πr^2 (since the top is closed)
Area of bottom = 0 (since the bottom is open)
Curved surface area = 2πrh
Total surface area = πr^2 + 2πrh
Now, we need to eliminate h from the expression for total surface area to get an expression in terms of r and x.
Volume of the cylinder = πr^2h
∴ h = V/(πr^2)
Substituting this value of h in the expression for total surface area, we get:
Total surface area = πr^2 + 2πr(V/(πr^2))
= πr^2 + 2V/r
Differentiating this expression with respect to r and equating it to zero to find the minimum surface area, we get:
d(Total surface area)/dr = 2πr - 2V/r^2 = 0
∴ r^3 = V/π
Let the thickness of the iron plate of the top be 5x/4.
Then, the radius of the top plate will be (r - 5x/4) and the height of the top plate will be x.
The volume of the top plate will be π(r - 5x/4)^2x
The total volume of the cylinder and the top plate will be:
V = πr^2h + π(r - 5x/4)^2x
Substituting the value of h from above, we get:
V = πr^2(V/(πr^2)) + π(r - 5x/4)^2x
Simplifying, we get:
V = πr^2/4(25x^2/16 - 4rx + 4r^2)
Differentiating this expression with respect to r and equating it to zero to find the minimum volume, we get:
dV/dr = πr/4(-8r + 25x/2) = 0
∴ r = 25x/16
Substituting this value of r in the expression for h, we get:
h = 16/(25π)x
Therefore, the ratio of the radius to the height of the cylinder using minimum material for the same capacity is:
r/h = (25x/16)/(16/(25π)x) = 25/4π
Simplifying, we get:
r/h = 4/5
Hence, option (C) is the correct answer.
Let the thickness of the plate forming the cylindrical sides be x.
Then, the thickness of the iron plate of the top will be 5/4x.
Let the radius of the cylinder be r and the height of the cylinder be h.
Volume of the cylinder = πr^2h
The minimum material required to make the cylinder will be when the total surface area is minimum.
Total surface area of the cylinder = Area of top + Area of bottom + Curved surface area
Area of top = πr^2 (since the top is closed)
Area of bottom = 0 (since the bottom is open)
Curved surface area = 2πrh
Total surface area = πr^2 + 2πrh
Now, we need to eliminate h from the expression for total surface area to get an expression in terms of r and x.
Volume of the cylinder = πr^2h
∴ h = V/(πr^2)
Substituting this value of h in the expression for total surface area, we get:
Total surface area = πr^2 + 2πr(V/(πr^2))
= πr^2 + 2V/r
Differentiating this expression with respect to r and equating it to zero to find the minimum surface area, we get:
d(Total surface area)/dr = 2πr - 2V/r^2 = 0
∴ r^3 = V/π
Let the thickness of the iron plate of the top be 5x/4.
Then, the radius of the top plate will be (r - 5x/4) and the height of the top plate will be x.
The volume of the top plate will be π(r - 5x/4)^2x
The total volume of the cylinder and the top plate will be:
V = πr^2h + π(r - 5x/4)^2x
Substituting the value of h from above, we get:
V = πr^2(V/(πr^2)) + π(r - 5x/4)^2x
Simplifying, we get:
V = πr^2/4(25x^2/16 - 4rx + 4r^2)
Differentiating this expression with respect to r and equating it to zero to find the minimum volume, we get:
dV/dr = πr/4(-8r + 25x/2) = 0
∴ r = 25x/16
Substituting this value of r in the expression for h, we get:
h = 16/(25π)x
Therefore, the ratio of the radius to the height of the cylinder using minimum material for the same capacity is:
r/h = (25x/16)/(16/(25π)x) = 25/4π
Simplifying, we get:
r/h = 4/5
Hence, option (C) is the correct answer.
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A cylindrical gas container is closed at the top and open at the bottom. If the iron plate of the top is 5/4 times as thick as the plate forming the cylindrical 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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