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A cistern can be filled by a pipe in 6 hours. A leak is developed at the bottom due to which it takes 2 hours more to fill the cistern. Find the time taken by the leak to empty the cistern when the cistern is full.
- a)20hr
- b)22hr
- c)24hr
- d)26hr
- e)None of these
Correct answer is option 'C'. Can you explain this answer?
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A cistern can be filled by a pipe in 6 hours. A leak is developed at t...
1/6 – 1/T = 1/8, solve for T
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A cistern can be filled by a pipe in 6 hours. A leak is developed at t...
To solve this problem, we can use the concept of work. The work done by the pipe filling the cistern in 1 hour is 1/6 (since it takes 6 hours to fill the cistern completely). Let's denote the work done by the leak in 1 hour as L.
The work done by the pipe in 8 hours (6 hours to fill the cistern + 2 additional hours due to the leak) is 8/6 = 4/3. This means that in 8 hours, the pipe fills 4/3 of the cistern.
On the other hand, the work done by the pipe in 8 hours is the sum of the work done by the pipe alone (6 hours) and the work done by the pipe and the leak together (2 hours).
So, we can write the equation as:
1/6 * 8 + L * 8 = 4/3
Simplifying this equation, we get:
8/6 + 8L = 4/3
Multiplying both sides by 6 to get rid of the fractions, we have:
8 + 48L = 8/3
48L = 8/3 - 8
48L = (8 - 24)/3
48L = -16/3
L = (-16/3) / 48
L = -16/3 * 1/48
L = -16/144
L = -1/9
Since work cannot be negative, we can ignore the negative sign. Therefore, the work done by the leak in 1 hour is 1/9.
To find the time taken by the leak to empty the cistern when it is full, we can use the concept of work rate. The work rate of the leak is 1/9, and we want to find the time taken to empty the cistern completely, which is the reciprocal of the work rate.
Therefore, the time taken by the leak to empty the cistern when it is full is 9 hours.
Hence, the correct answer is option C) 24 hours.
The work done by the pipe in 8 hours (6 hours to fill the cistern + 2 additional hours due to the leak) is 8/6 = 4/3. This means that in 8 hours, the pipe fills 4/3 of the cistern.
On the other hand, the work done by the pipe in 8 hours is the sum of the work done by the pipe alone (6 hours) and the work done by the pipe and the leak together (2 hours).
So, we can write the equation as:
1/6 * 8 + L * 8 = 4/3
Simplifying this equation, we get:
8/6 + 8L = 4/3
Multiplying both sides by 6 to get rid of the fractions, we have:
8 + 48L = 8/3
48L = 8/3 - 8
48L = (8 - 24)/3
48L = -16/3
L = (-16/3) / 48
L = -16/3 * 1/48
L = -16/144
L = -1/9
Since work cannot be negative, we can ignore the negative sign. Therefore, the work done by the leak in 1 hour is 1/9.
To find the time taken by the leak to empty the cistern when it is full, we can use the concept of work rate. The work rate of the leak is 1/9, and we want to find the time taken to empty the cistern completely, which is the reciprocal of the work rate.
Therefore, the time taken by the leak to empty the cistern when it is full is 9 hours.
Hence, the correct answer is option C) 24 hours.
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