Problem Solving

Joy Shaha wrote:

The 4 taps can fill the entire 500L tank in 12.5, and they are identical. Hence, each tap fills 125L every 12.5 hours, or 10L every one hour, or 5L every 30 minutes. The 4 taps together would fill 20L every 30 minutes.

The 5 emptying taps can completely drain the 500L tank in 20 hours, and they also are identical. Hence, each emptying tap drains 100L of water every 20 hours, or 5L every hour, or 2.5L every 30 minutes.

Now once the tank reaches 350L, one of the emptying taps is opened every 30 minutes.

Thus in the first 30 minutes: The 4 taps add a total of 20L, the one emptying tap removes 2.5L.

350L + 20L - 2.5L = 367.5L

In the next 30 minutes: The 4 taps again add 20L, with two emptying taps removing 5L.

367.5L + 20L - 5L = 382.5L

In the next 30 minutes: The 4 taps again add 20L, with 3 emptying taps removing 7.5L.

382.5L + 20L - 7.5L = 395L

In the next 30 minutes: The 4 taps again add 20L, with 4 emptying taps removing 10L.

395L + 20L - 10L = 405L

This is more than 400L, so we have gone too far. The final answer must be between 1.5 hours and 2 hours. Thus, the correct answer is 3. Note that 400L is directly in between the volume at 1.5 hours and the volume at 2.0 hours, and since the rate is constant over that time interval, the height would reach 400L exactly half way between (i.e. at 1.75 hours).

Let's use W = RT for this.

If we're filling the tank, W = 1, since we have one job to do. We're using 4 taps to do so, so R = 4*r, where r = the rate of each individual tap. The time is 12.5 hours, so T = 12.5.

1 = 4r * 12.5
r = 1/50 of the tank
r = 10 liters per minute

Now let's consider draining the tank.

1 = 5s * 20
s = 1/100 of the tank
s = 5 liters per minute

So we drain half as quickly as we fill.

Finally, let's get the tank from 350 to 400 liters.

We know W = RT, but our rate changes each half hour. As such, we're best off writing this as the sum of multiple rates. For instance, the first half hour has the rate 4r - s, but the second has 4r - 2s, etc. That gives us the equation

50 = .5*(4r - s) + .5*(4r - 2s) + .5*(4r - 3s) + ...

and we know the values of r and s, from above, so

50 = .5*(40 - 5) + .5*(40 - 10) + .5*(40 - 15) + ...

Notice that after three half hours, we're almost there (45 liters), so our time is between 1.5 and 2 hours. From here, either pick 1.75, or try adding .25*(40 - 20) to confirm that we've reached 50 liters.