An apple pie is taken out of an oven at a temperature of 200°F and placed on the counter
in a room where the temperature is 70°F. The temperature of the pie is 160°F after 15 min.
What is the temperature of the pie after 30 min?
Answers
Answer:
Explanation:
Newton's Law of Cooling states that the rate of cooling of an object is inversely proportional to the difference of temperatures between the object and its surroundings i.e.
d
T
d
t
=
−
k
T
, where
t
is the time taken and
T
is the difference of the temperatures between the object and its surroundings.
This gives us
T
as a function of
t
and is given by
T
(
t
)
=
c
e
−
k
t
.
With this it will take infinite time for object to cool down to room temperature, when
T
(
t
)
=
0
. Still let us assume that it cools to
72.5
o
F
or less, which is roundable to
72
o
F
and work it out.
Now as
T
(
0
)
=
c
e
−
k
×
0
=
c
=
375
−
72
=
303
and
T
(
15
)
=
303
×
e
−
15
k
=
375
o
F
−
215
o
F
=
160
o
F
or
e
−
15
k
=
160
303and
−
15
k
=
ln
(
160
303
)
=
−
0.638559
or
k
=
0.638559
15
=
0.0425706
If pie cools to
72.5
o
F
in
t
minutes, then
303
e
−
0.0425706
×
t
=
0.5
or
e
−
0.0425706
×
t
=
0.5
303
=
0.00165017
or
−
0.0425706
t
=
ln
0.00165017
=
−
6.40688
or
t
=
6.40688
0.0425706
=
150.5
Hence, it will take at lease
150.5
minutes to cool down to close to
72
o
F