How Long Does It Take to Fill a Sink with Both a Pipe and a Drain Open?

Imagine a scenario where a sink can be filled by a pipe in 5 minutes, but the same sink is drained in 7 minutes when the drain is open. You might wonder, how long would it take to fill the sink if both the pipe and the drain are open simultaneously?

This problem involves understanding the rates at which the pipe fills the sink and the drain empties it. By calculating these rates, we can determine the net rate at which the sink fills and, consequently, the time it takes to fill it under these conditions.

Finding the Filling Rates

The first step is to determine the filling rate of the pipe and the draining rate of the drain.

Filling Rate of the Pipe

The pipe can fill the sink in 5 minutes. Therefore, its filling rate is:

text{Filling Rate} frac{1 text{ tank}}{5 text{ minutes}} 0.2 text{ tanks per minute}

Draining Rate of the Drain

The drain can empty the sink in 7 minutes. Therefore, its draining rate is:

text{Draining Rate} frac{1 text{ tank}}{7 text{ minutes}} approx 0.1429 text{ tanks per minute}

Calculating the Net Rate

Next, we calculate the net rate at which the sink fills when both the pipe and the drain are open. This is achieved by subtracting the draining rate from the filling rate:

text{Net Rate} text{Filling Rate} - text{Draining Rate} 0.2 - 0.1429 approx 0.0571 text{ tanks per minute}

Time to Fill the Sink

Finally, we can find the time it takes to fill the sink when both the pipe and the drain are open by using the formula:

text{Time} frac{1 text{ tank}}{text{Net Rate}} frac{1}{0.0571} approx 17.5 text{ minutes}

Thus, it takes approximately 17.5 minutes to fill the sink when both the pipe and the drain are open.

Input-Output Analysis

Another way to look at the problem is through the input-output concept. The sink fills at a rate of 1/5 of the tank in a minute and empties at a rate of 1/7 of the tank in a minute. When both pipe and drain are open, the net rate of filling is:

frac{1}{5} - frac{1}{7} frac{7}{35} - frac{5}{35} frac{2}{35}

This means 2/35 of the tank is filled in one minute. Therefore, the time taken to fill the tank is:

frac{1}{frac{2}{35}} frac{35}{2} 17.5 text{ minutes}

Rephrasing the Solution

The pipe can fill the sink in 5 minutes, so 1/5 of the sink is filled in each minute. Similarly, the drain can empty the sink in 7 minutes, so 1/7 of the sink is drained in each minute. Therefore, the net amount of water added to the sink in one minute is:

frac{1}{5} - frac{1}{7} frac{7 - 5}{35} frac{2}{35}

This means that 2/35 of the sink is filled in one minute. To fill the entire sink, the time taken is:

frac{35}{2} 17.5 text{ minutes}

Thus, the time it takes to fill the sink with both the pipe and the drain open is 17.5 minutes.

Key takeaways:

Sink fill time: The time it takes to fill a sink. Pipe and drain: The mechanisms affecting the filling and draining of the sink. Rate of filling and draining: Understanding the rate at which the sink is filled and emptied.

Even though the different methods used may vary, the core concept and the final answer remain the same: it takes approximately 17.5 minutes to fill the sink with both the pipe and the drain open.