Question

a resorvoir can be filled by an inlet pipe in 24 hours and emptied by an outlet pipe 28 hours. the foreman starts to fill the resorvoir, but he forget to close the outlet pipe. six hours later he remembers and closes the outlet. how long does it take altogether to fill the reservoir? show your solution?

Answer 1

Let's find the rate that the inlet and outlet pipe can fill and empty the reservoirs.  remember that rate*time = work done so if we let the rate of the inlet pipe be i and the outlet pipe be 0 we have
i(24) = 1
o(28) = 1
if you're confused about what the 1 is, it is the number of reservoirs because in the problem, it gives us the time it takes for each pipe to either empty or fill 1 reservoir.  solving for r and t gives us:
i = 1/24 reservoirs/hour
o = 1/28 reservoirs/hour

in the first 6 hours the inlet pipe fills up (1/24)(6) = 1/4 reservoirs and the outlet pipe empties (1/28)(6) = 3/14 reservoirs so to find out how many reservoirs are filled we subtract emptied amount from filled amount:
1/4 - 3/14 = 1/28 reservoirs (note that if the amount emptied is greater than the amount filled you will obtain a negative answer. please just change that negative number to 0 because a negative answer simply means that emptying rate is greater than filling rate so you end up with no water).
now we need to figure out how long it will take to fill up 1 reservoir given we have already filled up 1/28 reservoirs and that the outlet pipe is finally closed.  to put it in simple terms, how long will it take for the inlet pipe to fill up the rest of the 27/28 of the reservoir.  good thing we've already found the rate the inlet pipe fills up reservoirs so we have the equation:
(1/24)t = 27/28
solving for t, we have 23.14 hours.  make sure you remember to add 6 to the answer because the question wants us to include that time in our answer.  doing so gives 29.14 hours.

let me know if you have any questions!

Answer 2

The flow rate of water into the reservoir is 1 reservoir per 24 hours. The flow rate out is one reservoir per 28 hours. For the six hours when the outlet pipe was open, the reservoir was filling at a rate of (1/24) - (1/28) = (7/168) - (6/168) = 1/168 reservoir per hour. Multiplied by 6 hours, this equals a total of 1/28 reservoir during the time the outlet pipe was left open.

The remaining volume of the reservoir will be filled at a rate of 1/24 reservoir per hour. What we need to do, then, is find the number of 1/24 reservoir portions in 27/28 reservoir. (27/28) = (162/168) = (23.142857/24). The entire reservoir, then will take 23.142857 + 6 = 29.142857 hours to fill.