Section 2.8 : Applications of Quadratic Equations
3. Two people can paint a house in 14 hours. Working individually one of the people takes 2 hours more than it takes the other person to paint the house. How long would it take each person working individually to paint the house?
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Start SolutionFirst, let Person A be the faster of the two painters and let t be the amount of time it takes to paint the house by himself. Next, let Person B be the slower of the two painters and so it will take this person \(t + 2\) hours to paint the house by himself.
Show Step 2Working together they can paint the house in 14 hours so we have the following word equation for them working together to paint the house.
\[\left( \begin{array}{c}{\mbox{Portion of job }}\\ {\mbox{ done by Person A}}\end{array} \right) + \left( \begin{array}{c}{\mbox{Portion of job}}\\ {\mbox{done by Person B}}\end{array} \right) = 1{\mbox{ Job}}\]We know that Portion of Job = Work Rate X Work Time so this gives the following word equation.
\[\begin{align*}\left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{of Person A}}\end{array} \right)\left( \begin{array}{c}{\mbox{Work Time}}\\ {\mbox{of Person A}}\end{array} \right) + \left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{of Person B}}\end{array} \right)\left( \begin{array}{c}{\mbox{Work Time}}\\ {\mbox{of Person B}}\end{array} \right) & = 1\\\left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{of Person A}}\end{array} \right)\left( {14} \right) + \left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{of Person B}}\end{array} \right)\left( {14} \right) & = 1\end{align*}\] Show Step 3Now we need the work rate of each person which we can get from their individual painting times as follows,
\[\begin{align*}\left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{of Person A}}\end{array} \right)\left( \begin{array}{c}{\mbox{Work Time}}\\ {\mbox{of Person A}}\end{array} \right) = \left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{of Person A}}\end{array} \right)\left( t \right) & = 1\\ & \Rightarrow \hspace{0.25in}{\mbox{Work Rate of Person A = }}\frac{1}{t}\end{align*}\] \[\begin{align*}\left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{of Person B}}\end{array} \right)\left( \begin{array}{c}{\mbox{Work Time}}\\ {\mbox{of Person B}}\end{array} \right) = \left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{of Person B}}\end{array} \right)\left( {t + 2} \right) & = 1 \\ & \Rightarrow \hspace{0.25in}{\mbox{Work Rate of Person B = }}\frac{1}{{t + 2}}\end{align*}\] Show Step 4Plugging these into the word equation from Step 2 we arrive at the following equation.
\[\begin{align*}\left( {\frac{1}{t}} \right)\left( {14} \right) + \left( {\frac{1}{{t + 2}}} \right)\left( {14} \right) & = 1\\ \frac{{14}}{t} + \frac{{14}}{{t + 2}} & = 1\end{align*}\] Show Step 5To solve this we know that we’ll need to multiply by the LCD, \(t\left( {t + 2} \right)\) in this case, to clear the denominators. Doing this gives,
\[\begin{align*}14\left( {t + 2} \right) + 14t & = t\left( {t + 2} \right)\\ 28t + 28 & = {t^2} + 2t\\ {t^2} - 26t - 28 & = 0\end{align*}\]After some simplification we arrive a fairly simple quadratic equation to solve. Using the quadratic formula gives,
\[L = \frac{{ - \left( { - 26} \right) \pm \sqrt {{{\left( { - 26} \right)}^2} - 4\left( 1 \right)\left( { - 28} \right)} }}{{2\left( 1 \right)}} = \frac{{26 \pm \sqrt {788} }}{2}\] Show Step 6Reducing the two values we got in the previous steps to decimals we arrive at the following two solutions to the quadratic equation from Step 2.
\[t = \frac{{26 - \sqrt {788} }}{2} = - 1.0357\hspace{0.25in}\hspace{0.25in}t = \frac{{26 + \sqrt {788} }}{2} = 27.0357\]The first solution to the equation doesn’t make any sense since it is negative (we are working with time and so it’s safe to assume we are starting at \(t = 0\) after all!) so that means the second is the answer we need.
This means that Person A can paint the house in 27.0357 hours while Person B can paint the house in 29.0357 hours (two hours more than Person A).