A thermometer is taken from a room where the temperature is 24oc to the outdoors, where the temperature is - 15oc. After one minute the thermometer reads 14oc. (a) what will the reading on the thermometer be after 3 more minutes?

Solution:

Use Newton's Law of Cooling.

T = T_s + (T_0 - T_s) * e^ (-kt)

where

T = temperature at any instant

T_s = temperature of surroundings

T_0 = original temperature

t = elapsed time

k = constant

Now, we need to find this constant. We are given that after one hour, the temperature drops to 13° C in a 7°C Environment.

T = 14, T_0 = 24, T_s = - 15, t = 1, k = ?

T = T_s + (T_0 - T_s) * e^ (-kt)

==> 14 = - 15 + (24 - 7) * e^ (-k)

==> 14 = 7 + 17*e^ (-k)

==> 7 = 17*e^ (-k)

==> 7/13 = e^ (-k)

==> - k = ln (7/17)

==> k = - ln (7/17) ≈ 0.774

Now,

Let's calculate temperatures!

T = ?, T_0 = 24, T_s = - 15, k = 0.773, t = 3

T = T_s + (T_0 - T_s) * e^ (-kt)

==> T = - 15 + (24 - (-15)) * e^[ - (0.774) (2) ]

==> T = - 15 + 39*e^ (-1.548)

==> T ≈ 15.72° C

This the required answer.