Science:Math Exam Resources/Courses/MATH152/April 2017/Question B 01 (a)

MATH152 April 2017

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Question B 01 (a)

Three engineering students, Xiuying, Arjan, and Marianne decide to take a summer job painting homes. Xiuying paints three times as fast as Marianne and twice as fast as Arjan and Marianne combined. All three together paint a room in four hours. (a) [2 marks] Let $x=(x_{1},x_{2},x_{3})^{T}$ be the vector, where $x_{1}$, $x_{2}$ and $x_{3}$ are respectively the number of rooms that Xiuying, Arjan, and Marianne can paint in an hour. Describe the information above as a linear system in the form $Ax=b$ (write $A$ and $b$ with specific values).

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Write each sentence describing how fast the engineers paint into linear equations with variables $x_{1}$, $x_{3}$, and $x_{3}$. To do this, note that how fast someone can paint can be re-phrased as the number of rooms that someone can paint in an hour. Afterwards, re-write the resulting system of linear equations into matrix form.

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Xiuying paints three times as fast as Marianne, so the number of rooms Xiuying can paint in an hour is three times the number of rooms Marianne can paint in an hour. In other words, $x_{1}=3x_{3}$. Xiuyin paints twice as fast as Arjan and Marianne combined, so the number of rooms Xiuying can paint in an hour is two times the number of room Arjan and Marianne together can paint in an hour. So, $x_{1}=2(x_{2}+x_{3})=2x_{2}+2x_{3}$. All three together paint a room in four hours, so in one hour all three together can paint one fourth of a room. Hence, $x_{1}+x_{2}+x_{3}=0.25$. The resulting system of equations is $x_{1}=3x_{3}$ $x_{1}=2x_{2}+2x_{3}$ $x_{1}+x_{2}+x_{3}=0.25$ Re-write the above system into the following $x_{1}+0x_{2}-3x_{3}=0$ $x_{1}-2x_{2}-2x_{3}=0$ $x_{1}+x_{2}+x_{3}=0.25$ In matrix form, this becomes $\color {blue}{\begin{pmatrix}1&0&-3\\1&-2&-2\\1&1&1\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}={\begin{pmatrix}0\\0\\0.25\end{pmatrix}}$

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