Question

During the summer the temperature inside the van reaches 55 degrees C while the outside is a constant 35 degrees C. When the driver gets in the van she turns on the AC which is set at 16 degrees C. If the time constant for the van is 1/k=2 hrs and the time constant for the van with AC is 1/k1=1/3 hr, when will the temperature inside the van reach 27 degrees C?

When I did this I got 30 minutes. The answer is 39.5 minutes. I cannot find my mistake.

Answer 1

Nevermind! Found my mistake. I input the wrong number for T.

Answer 2

Here is the work if anyone needs it.

M=35 T=55 T1=16 k=1/2 k1=3 ku=2.5

dT/dt=.5(55-T)+2.5(16-T)
dT/dt=67.5-3T=3(22.5-T)
integrate 1/(22.5-T) dT = 3 dt
-ln(22.5-T)=3t+c
22.5-T=ce^-3t
Input 55 for T
22.5-55=ce^0, c=-32.5
T=22.5+32.5e^-3t
To find when t is 27 degrees C:
27=22.5+32.5e^-3t
4.5=32.5e^-3t
.13846154=e^-3t
-ln(.13846154)/3=t
t=.65905423
To find the minutes:
(.65905423)(.6)=.39543=39.5 minutes
Hence, the temperature inside the van will reach 27 degrees C in 39.5 minutes.