Comprehension
These questions are based on the situation given below.
There are m blue vessels with known volumes \( v_1, v_2, \dots, v_m \) arranged in ascending order of volume, where \( v_1 = 0.5 \) litre and \( v_m < 1 \) litre. Each of these is full of water initially. The water from each of these is emptied into a minimum number of empty white vessels, each having volume 1 litre.
The water from a blue vessel is not emptied into a white vessel unless the white vessel has enough empty volume to hold all the water of the blue vessel. The number of white vessels required to empty all the blue vessels according to the above rule is \( n \).
Question: 1
Among the four values given below, which is the last upper bound on \( e \), where \( e \) is the total empty volume in the \( n \) white vessels at the end of the above process?
Among the four values given below, which is the last upper bound on \( e \), where \( e \) is the total empty volume in the \( n \) white vessels at the end of the above process?
Show Hint
In problems dealing with volume and upper bounds, carefully check the relationships between the variables to determine the correct upper bound.
Updated On: Aug 5, 2025
- \( m \cdot v_m \)
- \( m(1 - v_m) \)
- \( m \cdot v_1 \)
- \( m(1 - v_1) \)
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The Correct Option is B
Solution and Explanation
The total empty volume in the \( n \) white vessels is bounded by the formula \( m(1 - v_m) \), where \( v_m \) is the volume of the largest blue vessel. This represents the last upper bound for the total empty volume of the white vessels after the process is completed.
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Question: 2
Let the number of white vessels needed be \( n_1 \) for the emptying process described above, if the volume of each white vessel is 2 litres. Among the following values, which is the least upperbound on \( n_1 \)?
Let the number of white vessels needed be \( n_1 \) for the emptying process described above, if the volume of each white vessel is 2 litres. Among the following values, which is the least upperbound on \( n_1 \)?
Show Hint
When calculating upper bounds for vessels or containers, consider the total capacity required and use the smallest integer greater than or equal to the calculated number of vessels.
Updated On: Aug 5, 2025
- \( m/4 \)
- smallest integer greater than or equal to \( n/2 \)
- \( n \)
- greatest integer less than or equal to \( (n/2) \)
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The Correct Option is B
Solution and Explanation
To find the least upper bound on \( n_1 \), we observe that the number of white vessels required is bounded by the smallest integer greater than or equal to \( n/2 \), considering that each white vessel has a volume of 2 litres and we are distributing the water evenly.
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