Mod Exp Decay - im doin it wrong (1 Viewer)

[Starcraftmazter]

Newton's Law of Cooling states that when an object at temperature T degrees C is placed in an environment at temperature T0 C, the rate of temperature loss is given by the equation

dt/dt = k(T-T0)

where t is the time in seconds and k is a constant.

i) Show that T = T0 + A.e^kt is a solution to the equation.

ii) A packet of peas, initially at 24 Degrees C is placed in a snap-freeze refrigerator in which the initial temperature is maintained at -40 Degrees C. After 5 seconds the temperature of the packet is 19 Degrees C. How long will it take for the packet temperature to reduce to 0 Degrees C?

i) Easy stuff - 5 second job.

ii) Has me confused. The surrounding temp is -40 and the initial temp is 24 degrees, so the equation is:

T = -40 + 24.e^kt

At t=5, T=19, so working k out gives the result

k = ln (59/24)/5
Which is actually a positive number. But that shouldn't be, since this is decay, so k must be negative.
Otherwise it's impossible to work out how long it takes to get to 0 (with k being positive result, I got 2.8... seconds, which is obviously wrong).

So what gives? Why is k positive, what am I doing wrong?

 

[aalex]

T = -40 + Ae^kt
When t = 0, T = 24!!!, not A
24 = -40 + Ae^0
So A = 64
T = -40 + 64e^kt
When t = 5, T = 19, so k = ln(59/64)/5, which is negative.
Find t=?, for T =0.
t = 5ln(5/8)/ln(59/64)
t = 28.9 sec.

Good Luck!!!!!!!!:wave:

 

[Starcraftmazter]

Uhhh, *headbang, because the smiley isn't working*

Thanks.

 

[Legham]

My text books and shit say that the law of cooling is -k(T-T0) :/ Which would of explained the positive k, but if the question doesn't have a negative k then i guess you have to go by that :S

 

[Starcraftmazter]

That would be very interesting if in fact the equation was T = - 40 + 24.e^kt, and after so many seconds for it to be 19.

I can't imagine how that works =/

 

[Legham]

That would be very interesting if in fact the equation was T = - 40 + 24.e^kt, and after so many seconds for it to be 19.

I can't imagine how that works =/

Yeah, i didnt read the question. I just read the formula and what you were whinging about lol