a pool takes six hours to fill and eight hours to drain. The drain was accidently left open for the first three hours while the pool was filling and then closed. How long did it take for the pool to be filled?
fill rate = 1/6 pool/hr
drain rate = 1/8 pool per hour
rate with both = 1/6 -1/8 = 4/24 -3/24
= 1/24 pool/hour
so in three hours
(1/24 pool/hr )* 3 hr = 1/8 pool filled in first three hours
so we still need to fill 7/8 of pool
(1/6 pool/hr)* x hr = 7/8 pool
x = 42/8 hours to finish
so
3 + 42/8 = 8 1/4 hours total
To determine how long it took for the pool to be filled, we need to consider the rate at which the pool fills and drains.
Let's denote the rate at which the pool fills as "R fill" (meaning the amount of water added per unit of time) and the rate at which the pool drains as "R drain" (meaning the amount of water removed per unit of time).
From the given information, we know that it takes 6 hours to fill the pool completely. Therefore, the rate at which the pool fills is 1 pool/6 hours, which can be written as R fill = 1/6 pool/hour.
Similarly, we know that it takes 8 hours to drain the pool completely. Therefore, the rate at which the pool drains is 1 pool/8 hours, which can be written as R drain = 1/8 pool/hour.
Now, let's break down the time taken to fill the pool into different segments:
1. The drain was accidentally left open for the first 3 hours while the pool was filling. During this time, both filling and draining were happening simultaneously.
During the first 3 hours, the pool was being filled at a rate of 1/6 pool/hour. At the same time, the drain was removing water at a rate of 1/8 pool/hour.
Using the concept of rate × time = amount, we can calculate the amount of water that was added or removed during this period:
Amount filled = Rate × Time = (1/6 pool/hour) × (3 hours) = 1/2 pool
Amount drained = Rate × Time = (1/8 pool/hour) × (3 hours) = 3/8 pool
So, the net amount of water added during the first 3 hours is: (Amount filled) - (Amount drained) = (1/2 pool) - (3/8 pool) = 1/8 pool.
2. After the drain was closed, only the filling was happening. Since we have already filled 1/8 of the pool, we need to fill the remaining 7/8.
From the rate at which the pool fills (R fill = 1/6 pool/hour), we can determine the time needed to fill the remaining portion:
Time = Amount filled / Rate = (7/8 pool) / (1/6 pool/hour) = 7/8 × 6 = 42/8 = 21/4 hours
Therefore, it took an additional 21/4 hours (5 hours and 15 minutes) to fill the remaining 7/8 of the pool after the drain was closed.
To find the total time taken to fill the pool, we need to add the time taken during the first 3 hours (when both filling and draining were happening) and the time taken to fill the remaining 7/8 of the pool:
Total time = 3 hours + 21/4 hours = 3 hours + 5 hours and 15 minutes = 8 hours and 15 minutes.
Hence, it took 8 hours and 15 minutes for the pool to be filled after the drain was accidentally left open for the first 3 hours.