This is actually a question we went over in class, but I kind of spaced out when my teacher was explaining it. I have since solved it myself while I was reviewing for my calculus test, but when I compared my answer with the answer key my teacher provided to our class, I found out my answer was wrong. Here's the question.
"Write and solve the differential equation that models the statement in the following problem.
The rate of change of P with respect to t is proportional to 10-t."
I should add that this is part of our chapter involving exponential growth and decay.
The Attempt at a Solution
Now, I understand how to do this particular problem. You essentially start off with the following differential equation:
dP/dt = K/(10-t)
You then multiply the dt over to isolate all the t variable expressions on the right side.
dP = [K/(10-t)]dt
Then you would take an indefinite integral of both sides respectively. The problem is, my teacher says that the answer should be:
P = -K*ln|10-t| + C
When I solved this on my own, I produced the same result essentially, just without a negative sign in front of the K constant. Am I missing something here? Could someone please explain to me why there should be a negative sign or did my teacher make a mistake?