# Exponentials - Such a simple question yet...

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1. Hey everyone,

I know this is really simple and I cant work out why i keep getting it wrong, its simple exponentials but im literally driving myself insane over it.

The Question:

Newton's Law of cooling states that

T=Ta + (T0-Ta)e^-kt

Where T is the temperature at time t of an object, Ta is the ambient (room) temperature, T0 is the temperature at t=0, and k is a constant.

A dead environmentalist is found in a room with a thermostat maintaining the ambient temperature at 18 degrees C. At midnight (t=0) the temperature of the body is 26 degrees C. By 2am it has cooled to 22 degrees C. Find the value of k, sketch the graph of T as a function of time, and find the time of death, assuming a body temperature of 37 degrees C.

Now forgetting the sketching the graph bit of the question, i worked out k to be 5.776226505x10^-4?

Then using that to find the time of death i cant seem to get the right answer....

Cheers x
2. putting in the numbers....

22 = 18 + ( 26 - 18 ) e-2k

then rearrange to find k ?

37 = 18 + ( 26 - 18 ) e-kt

then rearrange to find t ( which will be negative... )
3. (Original post by the bear)
putting in the numbers....

22 = 18 + ( 26 - 18 ) e-2k

then rearrange to find k ?

37 = 18 + ( 26 - 18 ) e-kt

then rearrange to find t ( which will be negative... )
Yeah ive done that, but cant seem to get the right answer... ive got the exact same equation for finding k, except ive done it in seconds, so -1200k which gives me a value of 5.776226505x10^-4 for k.

I then plug that back into the same equation youve got (the second one) and it comes out as t=1497, which is only 25 mins? not near the 9000 seconds i need for the answer to come out as 9:30pm :L
4. (Original post by trm1)
Yeah ive done that, but cant seem to get the right answer... ive got the exact same equation for finding k, except ive done it in seconds, so -1200k which gives me a value of 5.776226505x10^-4 for k.

I then plug that back into the same equation youve got (the second one) and it comes out as t=1497, which is only 25 mins? not near the 9000 seconds i need for the answer to come out as 9:30pm :L
does the question tell you that t is seconds or minutes or hours ? ( i assumed it was hours )
5. (Original post by the bear)
does the question tell you that t is seconds or minutes or hours ? ( i assumed it was hours )
the answer just says 9:30pm, so its independent if you use t as seconds or hours. either t=2.5 if you use hrs, or t=9000 if you use seconds
6. (Original post by trm1)
Yeah ive done that, but cant seem to get the right answer... ive got the exact same equation for finding k, except ive done it in seconds, so -1200k which gives me a value of 5.776226505x10^-4 for k.

I then plug that back into the same equation youve got (the second one) and it comes out as t=1497, which is only 25 mins? not near the 9000 seconds i need for the answer to come out as 9:30pm :L
(Original post by the bear)
putting in the numbers....

22 = 18 + ( 26 - 18 ) e-2k

then rearrange to find k ?

37 = 18 + ( 26 - 18 ) e-kt

then rearrange to find t ( which will be negative... )
(Original post by Big Moisty)
uberteknik??
Be careful of how you interpret the two cases. It's easy to get confused so draw a quick sketch for each part and pay attention to T and T0 in particular.

NB It makes no difference to the final answer whichever time base is chosen save that the value for k will be different.

I have produced sketches so this becomes clearer:
Attached Images

7. (Original post by uberteknik)
Be careful of how you interpret the two cases. It's easy to get confused so draw a quick sketch for each part and pay attention to T and T0 in particular.

NB It makes no difference to the final answer whichever time base is chosen save that the value for k will be different.

I have produced sketches so this becomes clearer:
Thank you! after looking at that i realised my mistake was in calculating k... for some reason i was adamant that there are only 1200 seconds in two hours... so was plugging in -1200k. woops! thank you again though!
8. (Original post by trm1)
Thank you! after looking at that i realised my mistake was in calculating k... for some reason i was adamant that there are only 1200 seconds in two hours... so was plugging in -1200k. woops! thank you again though!
you are welcome

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