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Chapter 1: Problem 33

With both taps open, Robert can fill his kitchen sink in 5 min. When full, the sink drains in 10 min. How long will it take to fill the sink if Robert forgets to put in the stopper?

Short Answer

Expert verified
It will take 10 minutes to fill the sink.

Step by step solution

01

Determine the rates of filling and draining

First, find the rate of each tap and the drain. With both taps open, the sink fills in 5 minutes. So, the combined rate of both taps is \(\frac{1}{5}\) sink per minute. The sink drains in 10 minutes, so the rate of draining is \(\frac{1}{10}\) sink per minute.
02

Calculate the net rate of filling the sink

Subtract the drain rate from the fill rate to find the net rate of filling the sink. Thus, the net rate is: \[\frac{1}{5} - \frac{1}{10} = \frac{2}{10} - \frac{1}{10} = \frac{1}{10} \] sink per minute.
03

Find the time to fill the sink with the net rate

Knowing the net rate, determine the time it takes to fill the sink by taking the reciprocal of the net rate. So, the time to fill the sink is: \[\frac{1}{\frac{1}{10}} = 10 \] minutes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rates of Work
Understanding rates of work is crucial for solving many real-life problems. Rates of work tell us how much of a task can be completed in a specific time period. For example, if a sink can be filled in 5 minutes, the rate of filling is \(\frac{1}{5}\) sinks per minute. Similarly, if it drains in 10 minutes, the draining rate is \(\frac{1}{10}\) sinks per minute.
Sink Filling Problem
A common type of rate problem is the sink filling problem. Here, the goal is to determine how long it takes to fill or empty a sink given certain conditions. In the presented problem, Robert can fill his sink in 5 minutes with both taps running. If the sink drains when it is full, and it takes 10 minutes to drain, we need to calculate the effective or net rate to know the actual time to fill the sink.

We find the net rate by subtracting the draining rate from the filling rate: \[ \frac{1}{5} - \frac{1}{10} = \frac{2}{10} - \frac{1}{10} = \frac{1}{10} \] This result indicates the net rate at which the sink fills when the drain is open and both taps are running.
Time Calculation
To find how long it takes to fill the sink, we calculate the reciprocal of the net rate. The net rate, as found earlier, is \(\frac{1}{10}\) sinks per minute. The reciprocal of this rate gives time: \[ \frac{1}{\frac{1}{10}} = 10 \] minutes.

This means if Robert forgets to put the stopper in, it will take 10 minutes to fill the sink despite the draining because the net filling rate is slower.

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