Hey! I'm in AP Calculus in HS and can't figure out what I'm doing wrong in this problem.It's about the rate at which a coffe cup filled with coffee cools down.
Here it is:
Coffe Cup Problem
Data given:
D = Coffee temperature in F above room temp.
t = time in minutes
At t=0, D = 130
At t=3, D = 117
dD/dt = kD
D varies exponentially with t
Write the general solution for D. Use the given data to find the two constants in this particular solution.
This is what I've done:
dD/dt = kD
dD*D = k*dt Take integral of both sides
1/2D^2 = k*t
D^2 = 2kt
D^2 = 2*k*t
If I try to plug in t=0, D=130, I'll end up with 130 = 0 so obviously something went wrong... What did I do?
Suppose you pour a cup of coffee. When it is poured, it is at D = 130 degrees Fahrenheit above room temperature. Three minutes later it has cooled to 117 degrees F above room temperature. As the coffee cools the instantaneous rate of change of D with respect to time, t, in minutes is directly proportional to D.
If dD/dt is directly proportional to D, then D varies exponentionally with t. Use this fact to write the general solution for D. Use the given information to find the two constants in the particular solution.
It appears to me that you have made a trivial mistake:
1) dD/dt = kD
2) dD = kD dt ( divide the D out )
3) dD/D = k dt ( variables are isolated, and k is constant )
4) 1/D dD = k dt ( alternate form of #3 above )
5) integrate
5) ln|D| = kt ( the constant k comes out, dt = 1dt = t -> kt )
6) turn into exponential form
7) D = e^kt ( fundamental form of newton's law of cooling, where k is some constant of cooling and t is time )
This is newton's law of cooling. Go thenceforth from here, young padiwan :]
