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Pipes and cisterns

Pipes and cisterns is just another application of the concept of time and work. While we see only +ve work being done in normal cases of time and work, in case of pipes and cisterns, −ve work is also possible.

Given that pipes A and B can fill a tank in 20 minutes and 25 minutes working individually  this statement is similar to “A can do a work in 20 minutes and B can do the same work in 25 minutes”

Again, given that pipe C can empty a tank in 40 minutes we can say this statement is similar to “C can demolish a wall in 40 minutes (assuming that the work is building or demolishing the wall).

Let us understand this with the help of an example.
 
Example-1
A and B are two taps which can fill a tank individually in 10 minutes and 20 minutes respectively. However, there is a leakage at the bottom, which can empty a filled tank in 40 minutes. If the tank is empty initially, how much time will both the taps take to fill the tank (leakage is still there)?
Solution
Let us assume the units of work = LCM of (10, 20, 40) = 40 units
 
Work done by Tap A/minute = 4 units/minute (Positive work)
 
Work done by Tap B/minute = 2 units/minute (Positive work)
 
Work done by leakage/minute = 1 unit/minute (Negative work)
 
Net work done/minute = 5 units/minute
 
Hence, time taken = 8 minutes
 

Example-2
Pipe A can fill a tank in 3 hour. But there is a leakage also, due to which it takes 3.5 hour for the tank to be filled. How much time will the leakage take in emptying the tank if the tank is filled initially?
Solution
Assume the total units of work = 10.5 units
 
Work done by Tap A/h = 3.5 units/h (Positive work)
 
Work done by leakage/h = 3 units/h (Negative work)
 
Net work done/h = 0.5 units/h
 
So, the time taken = Description: 2318.png= 21 hour
 
Alternatively, due to the leakage, the pipe is required to work for an extra half an hour. So, the quantity filled by pipe in half an hour is being emptied by the leakage in 3.5 hour. Hence, the quantity filled by pipe in 3 hour will be emptied by the leakage in 21 hour.
 

Variable work

The concept of variable work comes from the possibility
 
⇒ that the rate of working can be different or
 
⇒ can be dependent upon some external agent.
 
In these cases, the rate of work will be proportional to some external factor.
 
Understand this with the help of a simple statement: The rate of the flow of water from a pipe is directly proportional to the area of the cross section of the pipe.
 
Example-3
There are three inlet taps whose diameters are 1 cm, 2 cm and 3 cm respectively. The rate of flow of the water is directly proportional to the square of the diameter. It takes 9 minutes for the smallest pipe to fill an empty tank. Find the time taken to fill an empty tank when all the three taps are opened.
Solution
The rate of flow of a diameter2, or, rate of flow = K × diameter2 (where K is a constant)
 
For 1st tap, rate of flow = K × 1
 
For 2nd tap, rate of flow = K × 4
 
For 3rd tap, rate of flow = K × 9
 
We know, the quantity filled will be equal to the product of the rate of flow and time.
 
So, the quantity filled by the smallest pipe = K × 1 × 9 = 9 K = Capacity of tank
 
Quantity of water filled by all the taps together in 1 minute = 9 K + 4 K + 1 K = 14 K
 
Assume that all the taps working together take ‘t’ minutes.
 
So, 14 K × t = 9 K
 
So, the time taken t = Description: 2327.png
 

Alternate work

The concept of alternate work is analogous to the concept of man-days. As we have seen in the concept of man-hour that if 20 men can do a work in 10 days, then this work is equivalent to 200 man-days. However, in the case of alternate work, two or more than two people of different efficiencies work alternately or in some particular pattern.
 
Example-4
Navneet can build a wall in 30 days and Rakesh can demolish the same wall in 40 days. If they work on alternate days with Navneet starting the job on the 1st days, then in how many days will the wall be built for the first time?
Solution
Let us assume the total units of work = 120 units
 
So, the wall built by Navneet in one day = 4 units
 
And wall demolished by Rakesh in one day = 3 units
 
So, effectively in two days, total wall built = 1 unit
 
Now, they work on alternate days, so days taken to built 116 units = 116 days
 
On 117th day Navneet will add another 4 units and so completing the construction of wall in 117 days.
 
(This problem can be understood well with another very traditional problem—A frog climbs up a pole 4 inches in 1 hour and slips 3 inches next hour. If height of the pole is 120 inches, then what is the time taken by the frog to reach the top of the pole?)
 




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