Science:Math Exam Resources/Courses/MATH102/December 2014/Question B 04

MATH102 December 2014

  •  QA 1  •  QA 2  •  QA 3  •  QA 4  •  QA 5  •  QA 6  •  QA 7  •  QA 8  •  QB 1  •  QB 2  •  QB 3  •  QB 4  •  QB 5  •  QB 6  •  QB 7  •  QC 1  •  QC 2(a)  •  QC 2(b)  •  QC 2(c)  •  QC 2(d)  •  QC 3  •  

   Other MATH102 Exams

  •  December 2014  •  December 2011  •  December 2012  •  December 2013  •  December 2016  •  December 2015  •  

Question B 04
The daily temperature in Vancouver during the month of March varies from an average low of $2^{\circ }$ C to an average high of $10^{\circ }$ C. It is coolest just before sunrise which is roughly 7 AM. Construct a function (using cosine) that provides a good description of the temperature $T(t)$ throughout a day in March where $t$ is measured in hours from 12 AM.

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
What is the general form of a cosine function? How do we stretch or shift a cosine (or sine) function?

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. The general expression of a cosine function is $T(t)=A+B\cos(\omega t+\varphi ).$ Suppose $A$, $B$ are both positive. Here the lowest temperature is $2^{o}$ C and highest is $10^{o}$ C, then $A-B=2,A+B=10,$ so $A=6$ and $B=4$. The value $2\pi /\omega$ is the period of this function. As it describes the temperature changes throughout a day which lasts for 24 hours, we have ${\frac {2\pi }{\omega }}=24,\quad \omega ={\frac {2\pi }{24}}.$ From the problem statement we know the lowest temperature is obtained at 7:00, by symmetry the highest is around 19:00. Then to shift the maxima from $0$ to $19$ we subtract $t$ by 19. So $\varphi =-19{\frac {2\pi }{24}}$ and the final function is $T(t)=6+4\cos \left({\frac {\pi }{12}}t-{\frac {\pi }{12}}19\right).$ Note the expression of the solution is not unique, it can be written in infinitely many ways, such as $T(t)=6+4\cos \left({\frac {\pi }{12}}t+{\frac {\pi }{12}}5\right),$