pipe can fill a cistern 8 hrs and another 10 hrs.both are opened at same time.if 2nd is closed 2 hrs before the cistern is filld up
pipe can fill a cistern 8 hrs and another pipe 10 hrs.both are opened at same time.if 2nd is closed 2 hrs before the cistern is filled up.find in what time,the cistern will be filled up?
rate of first pipe = 1/8
rate of 2nd pipe = 1/10
combined rate = 1/8 + 1/10 = 9/40
let the time taken to fill it be t hours
(t-2)(9/40) + 2(1/8) = 1 <----- the whole cistern
(9/40)t - 9/20 + 1/4 = 1
times 40
9t - 18 + 10 = 40
9t = 48
t = 48/9 hrs or 5 1/3 hrs
check:
3 1/3 hrs times 9/40 + 2 times 1/8
= (20/9)(9/40) + 2(1/8)
= 1
To find the time it takes for the cistern to be filled up when both pipes are opened at the same time and the second pipe is closed 2 hours before the cistern is filled up, we can use the concept of rates.
Let's assume that the first pipe can fill the cistern in 8 hours, which means it can fill 1/8th of the cistern in one hour. Similarly, let's assume that the second pipe can fill the cistern in 10 hours, which means it can fill 1/10th of the cistern in one hour.
When both pipes are opened at the same time, their combined rate of filling the cistern is the sum of their rates:
1/8 + 1/10 = 9/40
So, the cistern is filling up at a rate of 9/40th of its capacity per hour.
Now, we need to find the time it takes for the cistern to be filled up completely. Let's say it takes t hours.
From the information given, we know that the second pipe is closed 2 hours before the cistern is filled up. This means the first pipe works for t - 2 hours.
The amount of water filled by the first pipe in (t - 2) hours is:
(9/40) * (t - 2) = (9/40t) * (t - 2)
The amount of water filled by the second pipe in t hours is:
(1/10) * t = (1/10t) * t
The sum of these two amounts should be equal to the capacity of the cistern, which is 1 whole unit:
(9/40t) * (t - 2) + (1/10t) * t = 1
Now, we can solve this equation to find the value of t, which will give us the time it takes for the cistern to be filled up.
Simplifying the equation:
(9/40t) * (t - 2) + (1/10t) * t = 1
(9/40t) * (t - 2) + 1/10 = 1
(9/40t) * (t - 2) = 1 - 1/10
(9/40t) * (t - 2) = 10/10 - 1/10
(9/40t) * (t - 2) = 9/10
9(t - 2)/(40t) = 9/10
Cross multiplying:
9(t - 2) * 10 = 9 * 40t
90t - 180 = 360t
360t - 90t = 180
270t = 180
t = 180/270
t = 2/3
Therefore, it takes 2/3 of an hour (or approximately 40 minutes) for the cistern to be filled up when both pipes are opened at the same time and the second pipe is closed 2 hours before the cistern is filled up.