Question

An aluminum beam was brought from the outside cold into a machine shop where the temp. was held at 65 F. After 10 minutes the beam warmed up to 35 F and after another 10 minutes, its temp was 50 F. Use Newton's Law of cooling to estimate the beam's initial temp.

Answer 1

5° F

Explanation:

According to Newton's law of cooling, the rate of change is proportional to the difference between the temperature and the ambient temperature.

dT/dt = k (T − T₀)

Solving this by separating the variables:

dT / (T − T₀) = k dt

ln (T − T₀) = kt + C

T − T₀ = Ce^(kt)

T = T₀ + Ce^(kt)

We're given that T₀ = 65.

T = 65 + Ce^(kt)

At t = 10, T = 35.

35 = 65 + Ce^(10k)

-30 = Ce^(10k)

At t = 20, T = 50.

50 = 65 + Ce^(20k)

-15 = Ce^(20k)

Squaring the first equation:

900 = C² e^(20k)

Dividing by the second equation:

-60 = C

Therefore:

T = 65 − 60e^(kt)

At t = 0:

T = 65 − 60e^(0)

T = 5

The initial temperature is 5° F.