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Question A 04
The body temperature of an individual over a day is a rhythmic process. It reaches a maximum of 37.6 (degree Celsius) at 6:00 am and a minimum of 36.8 (degree Celsius) at 6:00 pm. Choose the trigonometric function that best describes , where is given in hours with taken at midnight (one minute after 11:59 pm).
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.
At 6:00 am, and at 6:00 pm . Translate the given information in terms of the given times.
Also, note that at max/min times the derivative of , , vanishes.
(Alternative solution) The trigonometric functions and are -periodic.
In other words, and
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
We need to check the following for each of the choices:
This means that choices (iii), (v) and (vi) are out since for all of them the second term vanishes and
The only option left is (iv). We check the conditions:
we can check the derivative too, but we don't really need.
(Alternative solution) By Hint 2, we know that and are -periodic.
This implies that for any constant and , both functions and are also -periodic;
Indeed, putting , we have .
Similar argument holds for .
However, since at (6 am) and at (6 pm), the function values of are different, i.e., ,
is not -periodic---(i), (ii), (iii) are out.
Then, using , given in (v) and (vi) satisfies , which is different from the given information ---(v), (vi) are out.