SOLUTION: Newton’s law of cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. So the rate of cooling f

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Question 1032283: Newton’s law of cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. So the rate of cooling for a bottle of lemonade at a room temperature of 75°F which is placed into a refrigerator with temperature of 38°F can be modeled by dT/dt=k(T-38) where T(t) is the temperature of the lemonade after t minutes and T(0) = 75. After 30 minutes the lemonade has cooled to 60°F, so T(30) = 60.
To the nearest minute, how long does it take for the lemonade to cool to 55°F?
Found 2 solutions by Fombitz, robertb:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Substitute,
u=T-38
du=dT
So,
du%2Fdt=ku
du%2Fu=kdt
ln%28u%29=kt%2BC
u=e%5E%28kt%2BC%29
u=Ce%5E%28kt%29
T-38=Ce%5E%28kt%29
So when t=0,
T=75
T-38=C%281%29
C=75-38
C=75
and
when t=30,
T=60
60-38=75e%5E%2830k%29
e%5E%2830k%29=%2860-38%29%2F75
e%5E%2830k%29=22%2F75
30k=ln%2822%2F75%29
k=%281%2F30%29ln%2822%2F75%29
So now the equation is complete.
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Find T when t=60
T-38=75e%5E%2860k%29
T-38=75e%5E%282%2Aln%2822%2F75%29%29%7D%7D%0D%0A%7B%7B%7BT-38=6.45%29
highlight%28T=44.5%29F
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Find t when T=55
55-38=75e%5E%28kt%29
e%5E%28kt%29=17%2F75
kt=ln%2817%2F75%29
%281%2F30%29ln%2822%2F75%29t=ln%2817%2F75%29
t=%2830ln%2817%2F75%29%29%2Fln%2822%2F75%29min
highlight%28t=36.3%29min

Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
The working equation given below is incorrect. It uses the equation T%28t%29+=38%2B75%2A%2822%2F75%29%5E%28t%2F30%29, which is flawed. It should be T%28t%29+=38%2B37%2A%2822%2F37%29%5E%28t%2F30%29, which actually takes into account the initial condition T(0) = 75.