Introduction:

Solving Differential Equations Symbolically and Newton's Law of Cooling:

On Thursday, we learned more about differential equations, this time using the symbolic (algebraic) approach. This topic tied into Newton's Law of Cooling. According to this law, a hot object cools at a rate proportional to the difference between its own temperature and that of its environment.

By applying this concept, we solved questions that dealt with cooling.

Content and Lessons:

First we began by deconstructing the equation of "dy/dx = ky." Mr. K mentioned in the past that normally, you are not allowed to pull apart the differential operator. However, without going through the complexities of why it is possible, we pulled it apart.

By seperating the two variables, including both 'dy' and 'dx', we found a familiar term that could be antidifferentiated.

Through algebraic play, we found the equation that defined the parent function of cooling.

*Note: the variable "c" and "C" are different in value but generally both represent a constant.

Next we looked at an example question.

- First we assigned variables, F = Temperature in Degrees of F and t = T time in Hours
- Second we applied the variables to "dy/dx=ky" to produce "dF/dt=k(F-20)" where 2o is the lowest temperature the object can attain in an environment
- By applying the same steps to seperate 'dF' and 'dt', we antidifferentiated and worked the out an equation that could be used to find 'F', the temperature, at any given time. The only unknown left is the constant that we have yet to solve.

At this point, we considerred both questions solved.

Next, we looked at another example question (Don't be overwhelmed by the length of the question as most of it is just a background story that Mr. K conjured up: The information required is located in paragraph 3).

By applying the same steps from the previous question, we found the solution to the problem. Again, our variable assignments were the same. The only difference is that the coldest temperature in this situation was 25 degrees. We found the equation that still included the constant value of C.

By taking this a step further, we inputted the coordinates given (temperatures at given times) and solved for the value of C to in turn, solve for the final equation.

Finally, by inputting the temperature of the cola at an unknown point in time, we solved for the uknown value of 't'.

Conlusion:

We went through two problems that dealt with Newton's Law of Cooling, solving differential equations symbolically.

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Homework is on the end of the slides posted in the previous post! Answer is on the following slide.

Tomorrow's Scribe is... Craig!

Happy Easter? Haha, good night.

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