Suppose a turkey is removed from an oven when it's inner temperature reaches 185°f and placed into a 75°f room to cool. the temperature in degrees farenheit can be modled by the function h=75+110(2.718281828)^-0.01277t, where t is measured in minutes. use the function to determine when the turkey will cool to 100°f.

We set H to 100 so we solve











Check:



~~~~~~

The above is what a good student might submit for this problem.  Let's go through it one more time, explaining it.

We set H to 100

We're told H is the temperature and to find when the temperature is 100, so we replace H by 100 in the equation and solve it.  We could have left the H in there and solved for t then substituted H.  I generally like it with letters better; who wants to write all those long numbers?  But here there were already constants in the equation so it made more sense to substitute first. 

so we solve  

I put in 100 for H and e for the 2.718...  I don't know if the student has transcribed the problem accurately but I suspect not.  I've noticed that whenever a student tries to help by changing the problem that's asked, that's usually where the student's problem is.  Yes e= 2.718... but math uses e a lot and the decimal value doesn't usually come up.  



I subtracted 75 for 100 and got 25.  I didn't need to explicitly write 100-25=75. That would count as showing my work, but I've showed plenty of work here.  A good studen can distinguish between showing their work and belaboring the obvious, turning a short problem into a novel.



I divided both sides by 110, which I think of more as moving the 110 to the other side as allowed by the rules.  Note I didn't bother to reduce the fraction. I think it's best to save the calculator to the end; get an exact answer first.



Here I took the natural logarithm of both sides.  That's how we expose the exponent, which we need do to ultimately solve for t.

Again I didn't evaluate the expression, I didn't change it to ln 25 - ln 110, I just left it.



Now I moved the -.01277 to the other side and have successfully solved for t.  Note the unevaluated expression has a particular merit that we must value highly.  It is the exact solution to the problem.

These sort of time constant problems always require an approximation at the end because that's the payoff.  How long? So now we get out the calculator. I prefer Wolfram Alpha because it's generally awesome.   The important thing to understand is both the exact answer and the approximation have worth.  I generally prefer not to soil my exact answer with an approximation, but apparently the world prefers otherwise.

Check:



The good student checks their own work.  I'm certain I make more many mistakes than 20 of you put together because I do so many problems. The only reason I can get consistently correct answers is that I check them.  I don't always include the check in the answer, but I almost always do them.  Sometimes I don't because I'm busy or lazy, and I often regret it.

That was a fun deconstruction of my own answer.  My impression is at least some students don't know some of the stuff I've mentioned above, so I hope it's helpful to somebody. Does anybody ever read these besides the person who asked the question? I've only been doing for a week or so I think and I really don't understand it.

The scale factor of what ?