Question

The temperature of a roast varies according to Newton's Law of Cooling: dT dt equals negative k times the quantity T minus A, where T is the water temperature, A is the room temperature, and k is a positive constant.
If a room temperature roast cools from 68°F to 25°F in 5 hours at freezer temperature of 20°F, how long (to the nearest hour) will it take the roast to cool to 21°F?

Answer 1

https://gyazo.com/23b71615e032fca0d654bb5437697556

Answer 2

@jim_thompson5910 Help always appreciated :)

Answer 3

were you able to find the T(t) function?

T = temperature
t = time

Answer 4

No

Answer 5

\[\Large \frac{dT}{dt} = -k(T-A)\]
\[\Large \frac{dT}{T-A} = -k dt\]
\[\Large \int\frac{dT}{T-A} = \int-k dt\]
I'll let you finish up with the integration. Tell me what you get

Answer 6

@jim_thompson5910 12?

Answer 7

how are you getting 12 ?

Answer 8

Would I integrate both sides?

Answer 9

yeah what do you get when you do so

Answer 10

I'm confused, how would I integrate the left side?

Answer 11

\[\Large \frac{dT}{dt} = -k(T-A)\]
\[\Large \frac{dT}{T-A} = -k dt\]
\[\Large \int\frac{dT}{T-A} = \int-k dt\]
\[\Large \ln(|T-A|) = -k*t + C\]
\[\Large |T-A| = e^{-k*t + C}\]
\[\Large |T-A| = e^{-k*t}*e^{C}\]
\[\Large |T-A| = e^{C}*e^{-k*t}\]
what comes next?

Answer 12

Having two equations, one positive and one negative.

Answer 13

T: roast temp (25 deg after 5 hrs)
A: freezer temp (20 deg)

so T-A is positive, so you can ignore the modulus sign

substitute T,A and t=5

Answer 14

you need to find k and c,
so you need another equation, so substitute t=0 and T=68

Answer 15

Final answer is 9