Science:Math Exam Resources/Courses/MATH100/December 2019/Question 09(a)
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Question 09(a) 

You put a bottle of room temperature water into your refrigerator. The temperature of your apartment is C, while the temperature of your refrigerator is C. After minutes the water has cooled to C. Give a formula for the temperature of the water minutes after it has been placed in the refrigerator. 
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 

This sort of problem is modelled by Newton's law of cooling. Let be the temperature of the water at time . Can you write down a relationship between , and possibly some constants? 
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.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Let be the temperature of the water minutes after putting it in the fridge. By Newton's law of cooling, the following equation holds
where is some constant and is the temperature of the surroundings. In our case, . Note that in the situation we're modelling, is larger than and the temperature is decreasing, so we should find that is negative. To solve such an equation, we may define the function , which has the same derivative as and therefore satisfies the following simpler equation
The solution to the above is
for some constant . We have then
The constants are determined by the other data of the question, which is the initial condition together with the data of the temperature at . The temperature of the apartment is the initial temperature of the water: . Thus
from which we conclude that . The other information in the question is . Thus
which is consistent with our comment above about . We may now write down the solution:
