Science:Math Exam Resources/Courses/MATH152/April 2010/Question B 01 (a)

MATH152 April 2010

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Question B 01 (a)

Three friends, Hartosh, Mark and Keiko decide to paint houses over the summer. Hartosh paints twice as fast as Mark. Hartosh and Keiko paint a home with six rooms in 8 hours. All three together paint a home with 14 rooms in 16 hours. (a) Let x = (x1, x2, x3)T be the vector of unknowns, where x1 is the number of rooms that Hartosh can paint in an hour, and x2 and x3 are respectively the number of rooms that Mark and Keiko can paint in an hour. Describe the information above as a linear system in the form $A\mathbf {x} =\mathbf {b}$ (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
There are three unknowns that form a 3 x 1 column vector. How many equations do I have for the three unknowns? Recall that the coefficients in front of each unknown make up the entries of the matrix A.

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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. Notice that we identify the speed that Hartosh can paint with as $x_{1}$, the speed that Mark can paint with as $x_{2}$, and the speed that Keiko can paint with as $x_{3}$ (all measured in rooms per hour). Our job is to translate each piece of information provided into a linear equation. Hartosh paints twice as fast as Mark This tells us that whatever Mark's speed is ($x_{2}$), Hartosh's speed ($x_{1}$) is twice that and therefore we get ${\begin{aligned}\displaystyle {}x_{1}=2x_{2}\end{aligned}}$ or, to write it like a linear equation with all three variables, ${\begin{aligned}\displaystyle {}x_{1}-2x_{2}+0x_{3}=0.\end{aligned}}$ Hartosh and Keiko paint a home with 6 rooms in 8 hours The total number of rooms that any person can paint in 8 hours is their speed multiplied by 8. Therefore the number of rooms that Hartosh, Mark, and Keiko paint in 8 hours is $8x_{1}$, $0x_{2}$ and $8x_{3}$ respectively where we notice that Mark painted 0 rooms because he wasn't involved in this particular project. The total number of rooms that get painted in that time is 6. Therefore, the sum of all the rooms that each person paints, must total to 6. With this in mind we have, ${\begin{aligned}\displaystyle {}8x_{1}+0x_{2}+8x_{3}=6.\end{aligned}}$ All three together paint a home with 14 rooms in 16 hours This is just like the last equation where Hartosh and Keiko painted the 6 rooms, but now all three people are contributing. Therefore, by similar logic to above we have ${\begin{aligned}\displaystyle {}16x_{1}+16x_{2}+16x_{3}=14.\end{aligned}}$ Now, we can summarize all this as the linear system ${\begin{aligned}x_{1}-2x_{2}+0x_{3}&=0\\8x_{1}+0x_{2}+8x_{3}&=6\\16x_{1}+16x_{2}+16x_{3}&=14.\end{aligned}}$ To write this as a matrix problem, we note that the coefficients in front of each variable form the matrix A and everything to the right of the equals sign, form the vector $\mathbf {b}$. Therefore, ${\begin{aligned}A={\begin{bmatrix}1&-2&0\\8&0&8\\16&16&16\end{bmatrix}}\end{aligned}}$ and ${\begin{aligned}\mathbf {b} ={\begin{bmatrix}0\\6\\14\end{bmatrix}}.\end{aligned}}$