According to Newton's Law of Cooling, the rate of the cooling is proportional to the temperature difference between the object and its surroundings.
In other words, if an object at a higher temperature say
T
0
is moved to a surrounding at a lower temperature
T
s
, the rate of cooling is directly proportional to the difference in temperature of te two i.e.
d
T
d
t
=
−
k
(
T
0
−
T
s
)
. This means that temperature
T
t
of the object at
t
time is given by
T
t
=
T
s
+
(
T
0
−
T
s
)
e
−
k
t
Here we have
T
0
=
23
∘
,
T
s
=
−
15
∘
and at
t
=
60
, we have
T
60
=
11
∘
- here we have used
60
as we intend to use time in seconds.
Hence
11
=
−
15
+
(
23
−
(
−
15
)
)
e
−
60
k
or
26
=
38
e
−
60
k
or
e
−
60
k
=
26
38
and hence
k
=
−
ln
(
26
38
)
60
=
0.00632482702
(a) After
5
more minutes i.e. after
360
seconds, temperature would be
T
360
=
−
15
+
38
e
−
360
×
0.00632482702
=
−
15
+
3.9
=
−
11.1
∘
(b) Let the temperature be
−
14
∘
C
after
t
seconds, then
−
14
=
−
15
+
38
e
−
0.00632482702
t
or
e
−
0.00632482702
t
=
15
−
14
38
=
0.026315789
i.e.
t
=
−
ln
0.026315789
0.00632482702
≅
575
Hence after
575
seconds i.e.
8
minutes and
35
seconds temperature would be
−
14
∘
C