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Water is filled up to a height h in a cylindrical vessel. It takes time t to completely drain the vessel by means of a small hole at the bottom. If water is filled up to a height 4h then the time it takes to completely drain the vessel is
- a)t
- b)4t
- c)2t
- d)t/4
Correct answer is option 'C'. Can you explain this answer?
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Water is filled up to a height h in a cylindrical vessel. It takes ti...
Given:
- Height of water filled in the cylindrical vessel = h
- Time taken to completely drain the vessel = t
To find:
- Time taken to completely drain the vessel when water is filled up to a height of 4h.
Solution:
When water is filled up to a height of h in the cylindrical vessel, the volume of water is given by the formula:
V = πr²h, where r is the radius of the vessel.
Step 1: Find the time taken to drain the vessel when it is filled up to a height of h:
Let's assume the cross-sectional area of the hole at the bottom of the vessel is A. The rate at which water flows out of the vessel is given by Torricelli's law:
v = √(2gh),
where g is the acceleration due to gravity.
The volume of water drained in time t is given by:
V = At.
Since the volume of water in the vessel is πr²h, we can equate the two expressions for volume and solve for A:
At = πr²h,
A = (πr²h) / t.
The rate of outflow of water is given by:
Q = Av = (πr²h / t) * √(2gh).
Since the volume of water in the vessel is πr²h, the time taken to drain the vessel is given by:
t = (πr²h) / (Q * √(2gh)).
Step 2: Find the time taken to drain the vessel when it is filled up to a height of 4h:
When the water is filled up to a height of 4h, the volume of water in the vessel is πr²(4h) = 4πr²h.
Using the same formula for the time taken to drain the vessel, we have:
t' = (πr²(4h)) / (Q * √(2g(4h))).
= 4((πr²h) / (Q * √(2gh))).
Comparing this with the expression for t, we can see that t' = 4t.
Therefore, the time taken to completely drain the vessel when water is filled up to a height of 4h is 4t. Thus, the correct answer is option (C).
- Height of water filled in the cylindrical vessel = h
- Time taken to completely drain the vessel = t
To find:
- Time taken to completely drain the vessel when water is filled up to a height of 4h.
Solution:
When water is filled up to a height of h in the cylindrical vessel, the volume of water is given by the formula:
V = πr²h, where r is the radius of the vessel.
Step 1: Find the time taken to drain the vessel when it is filled up to a height of h:
Let's assume the cross-sectional area of the hole at the bottom of the vessel is A. The rate at which water flows out of the vessel is given by Torricelli's law:
v = √(2gh),
where g is the acceleration due to gravity.
The volume of water drained in time t is given by:
V = At.
Since the volume of water in the vessel is πr²h, we can equate the two expressions for volume and solve for A:
At = πr²h,
A = (πr²h) / t.
The rate of outflow of water is given by:
Q = Av = (πr²h / t) * √(2gh).
Since the volume of water in the vessel is πr²h, the time taken to drain the vessel is given by:
t = (πr²h) / (Q * √(2gh)).
Step 2: Find the time taken to drain the vessel when it is filled up to a height of 4h:
When the water is filled up to a height of 4h, the volume of water in the vessel is πr²(4h) = 4πr²h.
Using the same formula for the time taken to drain the vessel, we have:
t' = (πr²(4h)) / (Q * √(2g(4h))).
= 4((πr²h) / (Q * √(2gh))).
Comparing this with the expression for t, we can see that t' = 4t.
Therefore, the time taken to completely drain the vessel when water is filled up to a height of 4h is 4t. Thus, the correct answer is option (C).
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Community Answer
Water is filled up to a height h in a cylindrical vessel. It takes ti...
Time required to empty the tank,
∴ t2 = 2t
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Water is filled up to a height h in a cylindrical vessel. It takes time t to completely drain the vessel by means of a small hole at the bottom. If water is filled up to a height 4h then the time it takes to completely drain the vessel isa)tb)4tc)2td)t/4Correct answer is option 'C'. Can you explain this answer?
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