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A cylindrical vessel filled with water upto a height H becomes empty in time t, due to a small hole at the bottom of vessel. If water is filled in same vessel to a height 4H, then time in which water level will reduce to height H, is
- a)4t
- b)3t
- c)t
- d)2t
Correct answer is option 'C'. Can you explain this answer?
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Solution:
Given:
- A cylindrical vessel filled with water up to height H becomes empty in time t, due to a small hole at the bottom of the vessel.
- Water is filled in the same vessel to a height 4H.
To find:
- The time in which the water level will reduce to height H.
Explanation:
1. Relationship between height and time:
- The time taken for the water level to reduce from 4H to H can be found by comparing the heights and times for the two cases.
- Let's assume that the initial height of the water in the first case is h₁ and the corresponding time is t₁.
- Similarly, let's assume that the initial height of the water in the second case is h₂ and the corresponding time is t₂.
2. Relationship between heights and time:
- The rate at which the water level decreases is directly proportional to the height of the water in the vessel.
- This implies that the time taken to reduce the height of the water by a certain fraction is the same, regardless of the initial height.
3. Applying the relationship to the given situation:
- In the first case, the water level reduces from h₁ to 0 in time t₁.
- In the second case, the water level reduces from h₂ to 0 in time t₂.
- Since the rate of decrease is the same, we can write the following proportion:
h₁ : h₂ = t₁ : t₂
4. Using the given information:
- In the first case, the water level reduces from H to 0 in time t.
- In the second case, the water level reduces from 4H to 0 in time t.
- Using the proportion, we can write:
H : 4H = t : t₂
5. Calculating the time to reduce the height to H:
- To find the time taken to reduce the height from 4H to H, we need to find t₂.
- Solving the proportion, we have:
H/4H = t/t₂
1/4 = t/t₂
t₂ = 4t
6. Conclusion:
- The time taken for the water level to reduce from 4H to H is 4t.
- Therefore, the correct answer is option C: 4t.
Given:
- A cylindrical vessel filled with water up to height H becomes empty in time t, due to a small hole at the bottom of the vessel.
- Water is filled in the same vessel to a height 4H.
To find:
- The time in which the water level will reduce to height H.
Explanation:
1. Relationship between height and time:
- The time taken for the water level to reduce from 4H to H can be found by comparing the heights and times for the two cases.
- Let's assume that the initial height of the water in the first case is h₁ and the corresponding time is t₁.
- Similarly, let's assume that the initial height of the water in the second case is h₂ and the corresponding time is t₂.
2. Relationship between heights and time:
- The rate at which the water level decreases is directly proportional to the height of the water in the vessel.
- This implies that the time taken to reduce the height of the water by a certain fraction is the same, regardless of the initial height.
3. Applying the relationship to the given situation:
- In the first case, the water level reduces from h₁ to 0 in time t₁.
- In the second case, the water level reduces from h₂ to 0 in time t₂.
- Since the rate of decrease is the same, we can write the following proportion:
h₁ : h₂ = t₁ : t₂
4. Using the given information:
- In the first case, the water level reduces from H to 0 in time t.
- In the second case, the water level reduces from 4H to 0 in time t.
- Using the proportion, we can write:
H : 4H = t : t₂
5. Calculating the time to reduce the height to H:
- To find the time taken to reduce the height from 4H to H, we need to find t₂.
- Solving the proportion, we have:
H/4H = t/t₂
1/4 = t/t₂
t₂ = 4t
6. Conclusion:
- The time taken for the water level to reduce from 4H to H is 4t.
- Therefore, the correct answer is option C: 4t.
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A cylindrical vessel filled with water upto a height H becomes empty in time t, due to a small hole at the bottom of vessel. If water is filled in same vessel to a height 4H, then time in which water level will reduce to height H, isa)4tb)3tc)td)2tCorrect answer is option 'C'. Can you explain this answer?
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