Question

A thermometer, reading 6°C, is brought into a room whose temperature is 22°C. One minute later the thermometer reading is 12°C. How long does it take until the reading is 20°C?

Physical information. Experiments show that the time rate of change of the temperature T of a body B (which conducts
heat well, for example, as a copper ball does) is proportional to the difference between T and the temperature of the
surrounding medium (Newton’s law of cooling).

a) Set up the model (the differential equation)
b) Find the general solution
c) Find the particular solution
d) Determine the constant of propotion
e) Find the time t.

Answer 1

a) Set up the model (the differential equation)

Newton's law of cooling states that the rate of change of the temperature T of a body B is proportional to the difference between T and the temperature of the surrounding medium. Mathematically, this can be expressed as:

dT/dt = -k(T - Tm)

where:

T is the temperature of the body at time t

Tm is the temperature of the surrounding medium

k is a constant of proportionality

In this case, the temperature of the body is the thermometer reading, which is initially 6°C. The temperature of the surrounding medium is 22°C. One minute later, the thermometer reading is 12°C. This means that k = 0.8.

b) Find the general solution

The general solution to the differential equation dT/dt = -k(T - Tm) is:

`T = C + Tm - (C - Tm)e^(-kt)`

where C is a constant of integration.

c) Find the particular solution

We can find the particular solution by substituting the initial conditions into the general solution. The initial conditions are:

T(0) = 6°C

Tm = 22°C

Substituting these values into the general solution, we get:

`6 = C + 22 - (C - 22)e^(-0.8*0)`

Solving for C, we get:

C = 14

Therefore, the particular solution is:

`T = 14 + 22 - (14 - 22)e^(-0.8t)`

d) Determine the constant of proportionality

The constant of proportionality k is already given. In this case, k = 0.8.

e) Find the time t

We want to find the time t when the thermometer reading is 20°C. Substituting this value into the particular solution, we get:

`20 = 14 + 22 - (14 - 22)e^(-0.8t)`

Solving for t, we get:

t ≈ 2.77 minutes

Therefore, it will take approximately 2.77 minutes for the thermometer reading to reach 20°C.