Hey ALAIN971.
This is a separable DE which means you move the T's to one side and the t's (and the differentials) to the other. Can you separate these variables and their differentials and integrate both sides?
A thermometer reading 70°F is placed in an oven preheated to
a constant temperature. Through a glass window in the oven
door, an observer records that the thermometer read 110°F after
.5 minute and 145°F after 1 minute. How hot is the oven?
using the following:
dT/dt = k(T- Tm)
Hey ALAIN971.
This is a separable DE which means you move the T's to one side and the t's (and the differentials) to the other. Can you separate these variables and their differentials and integrate both sides?
So now you have three equations for Tm, C and k.
70 = Tm + C.
145= Tm + Ce^(k)
110= Tm + Ce^(0.5k)
C = 70 - Tm
k = ln(145 - Tm) - ln(C)
k = 2*ln(110 - Tm) - 2*ln(C)
ln(145 - Tm) = 2*ln(110 - Tm) - ln(C)
145 - Tm = 70 + C
110 - Tm = 40 + C which means
ln(70 + C) = 2*ln(40 + C) - ln(C) which means
ln([70 + C]C) = ln([40 + C]^2) taking exponentials we get:
C[70+C] = [40+C]^2. and how you have a quadratic equation for C.