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### Section 2.3 : Applications of Linear Equations

5. A pump can empty a pool in 7 hours and a different pump can empty the same pool in 12 hours. How long does it take for both pumps working together to empty the pool?

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Start Solution

So, if we consider emptying the pool to be one job we have the following word equation describing both pumps working to empty the pool.

$\left( \begin{array}{c}{\mbox{Portion of job done}}\\ {\mbox{ by first pump}}\end{array} \right) + \left( \begin{array}{c}{\mbox{Portion of job done}}\\ {\mbox{ by second pump}}\end{array} \right) = 1{\mbox{ Job}}$

We know that Portion of Job = Work Rate X Work Time so this gives the following word equation.

$\left( \begin{array}{c}{\mbox{Work Rate of}}\\ {\mbox{ first pump}}\end{array} \right)\left( \begin{array}{c}{\mbox{ Work Time}}\\ {\mbox{of first pump}}\end{array} \right) + \left( \begin{array}{c}{\mbox{Work Rate of}}\\ {\mbox{ second pump}}\end{array} \right)\left( \begin{array}{c}{\mbox{ Work Time}}\\ {\mbox{of second pump}}\end{array} \right) = 1$ Show Step 2

We’ll need the work rates of each pump and for that we can use the information we have in the problem statement on each pump working individually and the following word equation for each pump doing the job individually.

$\left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{ of pump}}\end{array} \right)\left( \begin{array}{c}{\mbox{Work Time}}\\ {\mbox{ of pump}}\end{array} \right) = 1$

For the first pump we have,

$\left( \begin{array}{c}{\mbox{ Work Rate}}\\ {\mbox{of first pump}}\end{array} \right)\left( 7 \right) = 1\hspace{0.25in} \Rightarrow \hspace{0.25in}{\mbox{Work Rate of first pump = }}\frac{1}{7}$

and for the second pump we have,

$\left( \begin{array}{c}{\mbox{ Work Rate}}\\ {\mbox{of second pump}}\end{array} \right)\left( {12} \right) = 1\hspace{0.25in} \Rightarrow \hspace{0.25in} {\mbox{Work Rate of second pump = }}\frac{1}{{12}}$ Show Step 3

Now let $$t$$ be the amount of time it takes both pumps working together to empty the pool. Using this and the work rates we found in the second step our word equation from the first step becomes,

\begin{align*}\left( {\frac{1}{7}} \right)\left( t \right) + \left( {\frac{1}{{12}}} \right)t & = 1\\ \frac{{19}}{{84}}t & = 1\end{align*} Show Step 4

Now we can solve this for $$t$$.

$\frac{{19}}{{84}}t = 1\hspace{0.25in} \Rightarrow \hspace{0.25in}t = \frac{{84}}{{19}} = 4.4211$

So, it will take both pumps approximately 4.4211 hours to empty the pool if they both work together.