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Question 05 

FullSolution Problem. Justify your answer and show your work. Full simplification of numerical answers is required unless explicitly stated otherwise. A bottle of soda pop at room temperature (70°F) is placed in a refrigerator where the temperature is 40°F. After half an hour the soda pop has cooled to 60°F. How long does it take for the soda pop to cool to 50°F? 
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. 
Hint 1 

Newton's Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. (Try expressing this as a differential equation.) 
Hint 2 

Newton's Law of Cooling can be expressed as , where is the temperature of an object, is the temperature of its surroundings, and is some constant. What differential equation does this remind you of? What is the general solution of equations in this form? 
Hint 3 

Try making the substitution to obtain the equation . 
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.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. The solution of the differential equation is
